Would the wedding be performed by a priest (kohen)? The community's elders? The couple's parents? Perhaps nobody would have officiated it and the couple would just publicly agree to be married?
What would the ceremony be like? I would guess the wedding ring is just Raphael projecting back wedding practices from his own era.
Also, who would a 16th century Italian painter think would have officiated the wedding of Mary? Does Raphael think he's painting a Jewish priest? A rabbi?
There is a ton of anachronism in that painting.
Just within the Talmud itself there is significant discussion about every detail of marriage (likewise divorce). The first textual evidence I'm aware of, as a religious Jew with textual familiarity, is from the Mishna which was compiled by Judah The Prince (Yehuda HaNasi) sometime in the 300s CE. He took the oral tradition, which is significantly older and records many legal rulings by name and place, and made it a legible record.
An example is a discussion between the followers of Hillel (Beit Hillel) and Shammai (Beit Shammai). Those guys each had their own legal ruling that came out somewhere between the last days of the BCE and the first century BC. This was in the time of the Roman occupation and the Roman-Jewish wars.
The very first "mishna" of tractate Kiddushin(Marriage) states that there are 3 ways to marry and 2 ways to divorce. This is the oldest statement we have textual evidence stating a practice. Those ways are Kesef (Money), Shtar (Contract), and Bi'ah (Relations).
From there it discusses other side issues like how much money, what kind of contract, etc. which later in Jewish law extrapolates further as time and history go on.
It's entirely possible practice differed, but, the known tradition has always involved an exchange of money or contract (relations was nixed as an improper way to go about things but the final deal sealer for all 3 options due to the 2-part nature of engagement/marriage).
You would've needed witnesses there who aren't related to either side or each other, and a contract anyways would have been drawn up to state the terms of the marriage and what would happen in case of dissolution.
A ring, itself, is a relatively modern expression of the money side. There are textual references to all sorts of things....so..in theory, if the receiving side(bride's family) were down with goats, you could use goats.
The idea works on the more ancient concept of money--that is, something of value that people willingly accept.
This isn't really a historian statement on my part, but, more of a nuanced understanding based on Jewish practice in history as a member of that tradition.