The Peirush Baalei HaTosfos says that Yehuda gave Tamar his ring because he married her with it. In that he was a person of high status, he certainly had at least two people accompanying him, and they served as witnesses as he married her by giving her his ring. She remained anonymous to him, as we see from the search for "the Kedeisha at the crossroad," but you don't have to know a woman's name to marry her, especially when you don't plan to see her after the first night.
The idea that Yehuda was mekadeish Tamar is not unique to the Baalei HaTosfos; the Re'eim says this as well, and there's no reason to think it's inconsistent with the way the story is presented. Even if she appeared to be a kedeisha, there was no reason Yehuda couldn't kasher the event through kiddushin. What is surprising is the idea that Yehuda gave her the Chosemes because he wanted to be mekadeish her with a ring. This is an anachronism. The now universal minhag to do kiddushin with a ring is not mentioned anywhere in the Gemara or Medrashim other than the Zohar, as brought in the Rama EH 27. The first general reference to this is in the Teshuvos of the Geonim. So if you see it to mean that he gave her his signet ring somewhat like you give someone your driver's license to ensure that you'll come back, that would be fine. But he learns that he gave her the ring because that's how people do kiddushin.
Of course, like all minhagim, everyone has an explanation and a holy symbology for using a ring. They are all beautiful, and most of them are about as meaningful as the explanations for kreplach, but I want to point out that according to the Rogotchover's explanation, we can understand why Yehuda davka wanted to be mekadeish her with a ring, two thousand years before it became the common practice.
The Rogotchover offers a lomdishe reason for the minhag: the kiddushin has to be effected by an unrestricted gift. If the man retains any rights in the object, the kiddushin is not valid. At the time of the Mishna and Gemara, Kiddushin took place long before Nisuin, but now the two are simultaneous, because the kiddushin takes place beneath the chuppah (and according to some, the chupa actually precedes the kiddushin, in that the badeken is the chuppah; see EH 27.) When the woman is married, her husband acquires the right to use her property and to keep the profits from that use- he has a kinyan peiros. So when Kiddushin was long before Nisuin, there was no problem He would give her an object of value, and it was totally hers, at least until the nisuin. But now that the two are simultaneous, when he gives her the object, and she becomes his wife, he immediately acquires the right to use and invest it, and it cannot be said to be entirely hers. The only way to avoid this problem is to give her something that he has no rights in- clothing or jewelry. So the minhag of giving jewelry was encouraged by the Rabbanim.
Here is the Rogotchover on the Rambam Ishus 3:1
So according to the Rogotchover, in Yehuda's case, it was necessary that he give her a ring- namely, a tachshit- for kiddushin, because the kiddushin and nisuin were simultaneous.
It's funny that Hagaon haRav Shernbuch brings this Baalei Hatosfos, and he brings the Rogotchover, but אגב ריהטא he doesn't make the connection- he didn't notice that the latter explains the former.
UPDATE:
Over Shabbos, I realized that the joke might be on me. The din peiros is, obviously, a derabanan, and there is no reason to assume it existed at the time of Yehuda and Tamar.
But I want to remind you that we're saying pshat in the shitta of the Baalei Tosfos. Allow me to direct you attention to Tosfos in Gittin 47b, to wit-
Since I mention the minhag to use a ring, here is a list of the reasons I've seen- good, bad, and indifferent- that people use to give meaning to the minhag. Someone might find this useful. As I said above, I personally think that most of them are on par with the reasons people give for eating kreplach.
1. There is one passuk in which the letter Mem in middle of a word is formed like a mem sofis, an end-of-word Mem, the passuk in Yeshaya 9:7-לםרבה המשרה ולשלום אין קץ. The Tikunei Zohar mentioned in the Rama in 27 says this letter Mem is ring shaped, and it alludes to the peace and fertility we portend for this couple. The Gematria of לםרבה is Eizer and Zera. This Zohar is also brought more clearly in the Levush here in siman 10.
2. The Chinuch in 552, it is a reminder to her and to all that she is a married woman. One assumes that she doesn't need the reminder on Shabbos, since the halacha used to be that it was assur to wear jewelry on Shabbos. Unless this doesn't have a din of jewelry, which I doubt.
3. In next week's parsha we will see that Pharaoh gave Yosef his ring as a sign of trust and high office and responsibility. We find the same with Achashveirosh and Haman. So, too, when a woman becomes the Akeres HaBayis, she receives the ring of honor, responsibility, and trust. אוצר כל מנהגי ישרון
4. A ring joins two objects, as do the taba'os of the Krashim of the Mishkan and of the Choshen and Eifod. This ring joins the husband and wife. לקוטי מהרי"ח
5. A chain is made up of rings. The creation of a new family forges another link in the chain of mesora of Klal Yisrael, the Shalsheles HaYuchsin. ספר המטעמים אות לא
6. Through marriage, the Shechina is shoreh in a house. The ring is a symbol of the Shechina because it has no beginning and no end.
7. The Rogotchover I mentioned above, to avoid the kinyan peiros problem where the kiddushin and the chuppah are simultaneous.
8. I'm not sure this counts as a reason, but the Mordechai in Kiddushin 488 says that a 14 K gold ring is something that doesn't need evaluation, it has inherent value according to its weight. (In fact, some communities use silver rings; it is possible that this is based on the din of כספא טיבעא.) Perhaps that's why a gold ring is used, because clothing or gems or things need assessment. Even coins used to vary according to whether they were used locally as currency. The value of a simple piece of gold is pretty universal, the same all around the world, except, of course, in places of unusual circumstances like war or famine. The Mordechai- העם נהגו לקדש בטבעת של זהב בלא אבן יקרה בתורה משום דרוב בני אדם אינם בקיאים בשומא
9. , .טבעת בגימטריה קידושין
Not exactly, Taba'as is 481 and Kiddushin is 480.
Please note that the only one that addresses the fact that rings were not used until after Chasimas HaShas is the Rogotchover.
Honestly, there is no need to explain the minhag. It's ancient and almost universal among races, religions, and cultures (the Puritans tried to stop it and gave the bride a thimble, but many of them cut off the top and wore it as a ring anyway) with the exceptions of China and Japan. Still, we like to see significance in the things we do, so that's fine. As they say, מנהג ישראל תורה. The explanations remind me of the תפארת ישראל in פסחים פרק י' בועז אות ג that says
Why do I say it's not a specifically Jewish thing? Maybe we thought of it first! The answer is no, because
1. That's just silly.
2. We find plenty of historical evidence of this use among Gentiles long before there's any record of its use among the Jews.
3. All these explanations only deal with the wedding band that we use for Kiddushin, and do not address the engagement ring, which is now universal among the Jews as well. Even the Satmarer's takana only replaced the diamond with an artificial stone, (and here) but did not do away with the minhag to give a girl an engagement ring.
ותאמר חותמך וגו', פירוש טבעתך שאתה חותם בה וקדשה בטבעת. ותימא לה"ר משה וכי מה הועילו הקידושין, והרי צריכים עדים, ואפילו שניהם מודים כדאיתא במסכת קדושין ויש לומר דמסתמא אדם חשוב כיהודה לא היה יוצא לדרך בלא שנים, רעהו ואחר עמו, כמו תלמיד חכם, וקדשה בפניהם. ואם תאמר ואכתי היאך הועילו הקדושין,והלא לא מסר לה הטבעת אלא בתורת משכון, ואמרינן בקדושין )דף ח.( קדשה בפרוטה והניח משכון עליה אינה מקודשת. ויש לומר שהקנה לה הטבעת באותה שעה לגמרי, אמנם לאחר מכן התנה עמה שאם שולח גדי עזים שתחזיר לו הטבעת.
Of course, like all minhagim, everyone has an explanation and a holy symbology for using a ring. They are all beautiful, and most of them are about as meaningful as the explanations for kreplach, but I want to point out that according to the Rogotchover's explanation, we can understand why Yehuda davka wanted to be mekadeish her with a ring, two thousand years before it became the common practice.
The Rogotchover offers a lomdishe reason for the minhag: the kiddushin has to be effected by an unrestricted gift. If the man retains any rights in the object, the kiddushin is not valid. At the time of the Mishna and Gemara, Kiddushin took place long before Nisuin, but now the two are simultaneous, because the kiddushin takes place beneath the chuppah (and according to some, the chupa actually precedes the kiddushin, in that the badeken is the chuppah; see EH 27.) When the woman is married, her husband acquires the right to use her property and to keep the profits from that use- he has a kinyan peiros. So when Kiddushin was long before Nisuin, there was no problem He would give her an object of value, and it was totally hers, at least until the nisuin. But now that the two are simultaneous, when he gives her the object, and she becomes his wife, he immediately acquires the right to use and invest it, and it cannot be said to be entirely hers. The only way to avoid this problem is to give her something that he has no rights in- clothing or jewelry. So the minhag of giving jewelry was encouraged by the Rabbanim.
Here is the Rogotchover on the Rambam Ishus 3:1
אם בכסף הוא מקדש. והמנהג שמקדש בטבעת ולא בכסף. ונ'ל דהנה עי׳ בירושלמי פרק ה׳ דנדרים דהמקדש בספר תורה שלו אינה מקודשת וטעם כמ״ש הר"ן פרק א׳ דקדושין גבי מנה אין כאן דבדבר שלא יצא מידו לגמרי לא הוה קדושין והנה מבואר בב״ב ד׳ מ״ג דמן ס״ת אינו יכול לסלק לגמרי יע״ש ד׳ קנ״א ע״א גבי אם מיקרי נכסו ואף דהא בגיטין דף כ׳ אמרינן דאם גירשה בס״ת אם אקדים זוזא לספרא כו׳ הוה שייך גט התם כיון שנכתב לשם גט נפסלה הס"ת אבל בס״ת כשירה אינה
מקודשת והנה כעת המנהג שמקדשים בשעת חופה ותיכף הוה נשואה וא״כ יהא לו בכסף קדושין פירות וא״כ לא הוה קדושין לכך תתקדש בכלי דהוה כלבוש ועי׳ כתובות ר' נ״ד ע״א וסנהדרין ל׳ קי״ב ומש" כ רבינו ז״ל בפרק א׳ מהלכות מלוה ה״ה וכ״מ בזה ועי׳ ירושלמי נזיר פרק ה׳ הל׳ ב׳ דאין בע״ח גובה מתכשיטין של אשתו
So according to the Rogotchover, in Yehuda's case, it was necessary that he give her a ring- namely, a tachshit- for kiddushin, because the kiddushin and nisuin were simultaneous.
It's funny that Hagaon haRav Shernbuch brings this Baalei Hatosfos, and he brings the Rogotchover, but אגב ריהטא he doesn't make the connection- he didn't notice that the latter explains the former.
UPDATE:
Over Shabbos, I realized that the joke might be on me. The din peiros is, obviously, a derabanan, and there is no reason to assume it existed at the time of Yehuda and Tamar.
But I want to remind you that we're saying pshat in the shitta of the Baalei Tosfos. Allow me to direct you attention to Tosfos in Gittin 47b, to wit-
ולביתך מלמד שאדם מביא בכורי אשתו. וא"ת והא מן התורה אין לבעל פירות בנכסי אשתו אלא תקנתא דרבנן בעלמא היא א"כ ע"כ גזירת הכתוב הוא ויש לומר דרך נשים שנותנות פירות לבעליהן וכי האי גוונא איירי קרא וכענין זה אמרינן בריש קידושין (דף ד.) אי כתב רחמנא מעשה ידיה לאביה דקא מיתזנא מיניה ואע"ג דמדאורייתא אינו חייב במזונות בתו אלא דאורחא דמילתא הוא שהוא זן אותה:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Since I mention the minhag to use a ring, here is a list of the reasons I've seen- good, bad, and indifferent- that people use to give meaning to the minhag. Someone might find this useful. As I said above, I personally think that most of them are on par with the reasons people give for eating kreplach.
1. There is one passuk in which the letter Mem in middle of a word is formed like a mem sofis, an end-of-word Mem, the passuk in Yeshaya 9:7-לםרבה המשרה ולשלום אין קץ. The Tikunei Zohar mentioned in the Rama in 27 says this letter Mem is ring shaped, and it alludes to the peace and fertility we portend for this couple. The Gematria of לםרבה is Eizer and Zera. This Zohar is also brought more clearly in the Levush here in siman 10.
2. The Chinuch in 552, it is a reminder to her and to all that she is a married woman. One assumes that she doesn't need the reminder on Shabbos, since the halacha used to be that it was assur to wear jewelry on Shabbos. Unless this doesn't have a din of jewelry, which I doubt.
3. In next week's parsha we will see that Pharaoh gave Yosef his ring as a sign of trust and high office and responsibility. We find the same with Achashveirosh and Haman. So, too, when a woman becomes the Akeres HaBayis, she receives the ring of honor, responsibility, and trust. אוצר כל מנהגי ישרון
4. A ring joins two objects, as do the taba'os of the Krashim of the Mishkan and of the Choshen and Eifod. This ring joins the husband and wife. לקוטי מהרי"ח
5. A chain is made up of rings. The creation of a new family forges another link in the chain of mesora of Klal Yisrael, the Shalsheles HaYuchsin. ספר המטעמים אות לא
6. Through marriage, the Shechina is shoreh in a house. The ring is a symbol of the Shechina because it has no beginning and no end.
7. The Rogotchover I mentioned above, to avoid the kinyan peiros problem where the kiddushin and the chuppah are simultaneous.
8. I'm not sure this counts as a reason, but the Mordechai in Kiddushin 488 says that a 14 K gold ring is something that doesn't need evaluation, it has inherent value according to its weight. (In fact, some communities use silver rings; it is possible that this is based on the din of כספא טיבעא.) Perhaps that's why a gold ring is used, because clothing or gems or things need assessment. Even coins used to vary according to whether they were used locally as currency. The value of a simple piece of gold is pretty universal, the same all around the world, except, of course, in places of unusual circumstances like war or famine. The Mordechai- העם נהגו לקדש בטבעת של זהב בלא אבן יקרה בתורה משום דרוב בני אדם אינם בקיאים בשומא
9. , .טבעת בגימטריה קידושין
Not exactly, Taba'as is 481 and Kiddushin is 480.
Please note that the only one that addresses the fact that rings were not used until after Chasimas HaShas is the Rogotchover.
Honestly, there is no need to explain the minhag. It's ancient and almost universal among races, religions, and cultures (the Puritans tried to stop it and gave the bride a thimble, but many of them cut off the top and wore it as a ring anyway) with the exceptions of China and Japan. Still, we like to see significance in the things we do, so that's fine. As they say, מנהג ישראל תורה. The explanations remind me of the תפארת ישראל in פסחים פרק י' בועז אות ג that says
כשקבלו מלה יוונית לתוך לשוננו הקדושה הכריכוה להתיהד תחילה לכן שבשו את המלה במקורה בכוונה והסבו לקרותה באופן אחר דהיינו לפי הנאות לפרשה ע"פ דרך לשונו הקדושה
Why do I say it's not a specifically Jewish thing? Maybe we thought of it first! The answer is no, because
1. That's just silly.
2. We find plenty of historical evidence of this use among Gentiles long before there's any record of its use among the Jews.
3. All these explanations only deal with the wedding band that we use for Kiddushin, and do not address the engagement ring, which is now universal among the Jews as well. Even the Satmarer's takana only replaced the diamond with an artificial stone, (and here) but did not do away with the minhag to give a girl an engagement ring.