Sam2:
And Rabbi Linzer (my Rav says that he is still Frum and worthy of the title) admits that it doesn’t mean anything. The whole point of the double-ring ceremony is to make the woman feel good about herself. He even says that you have to explain this because he points out some problems in the Kiddushin if they think that she needs to give a ring as well.
R’ Moshe in Even Ha’ezer 3:18 writes that the Kiddushin is perfectly valid. Although he does hold that it’s assur to do it.