Do you still do traditional marriages with a penny?
Marriages can be done with a penny, but the tradition is to perform a marriage with a wedding ring, and not with coins. In a technical (halachic/Jewish Law) sense, there is no difference between the two.
Chinuch (552) explains why a ring is used: “We may also say that it is in order that the bride should forever recall that she is married to this man, and she will never betray him.… Thus they will dwell in peace in the world. Because of this principle that I have stated, the custom of Israel is to perform the kiddushin with a ring.”
We may add that while most possessions owned by husband or wife end up belonging to the “family” and are enjoyed by all members of the household, using a ring to transact the marriage assures that the item will belong exclusively to the wife.