In the gemara today, there is a hava amina that kiddushin can be accomplished by her giving money to him and/or her saying she is mekadesh her (or that she is mikudeshes to him). R' Moshe in Dibros Moshe points out that this hava amina seems to undermine the entire process of kiddushin. Kiddushin is for him to acquire her, and it is strange for the gemara to entertain that she can acquire him. But he explains based on rashi that the gemara thinks that perhaps kiddushin is not a kinyan at all, rather the exchange of money from him to her or even her to him creates a commitment from both of them to join in a state of marriage. The gemara rejects this notion and insists the kiddushin is only when he "acquires" her.
R' Moshe has 2 teshuvos regarding a double ring ceremony. In (E.H. 3:18) he explains that if he gives her a ring and says "הרי את מקודשת לי" and then she does the same, although it is assur [based on a brilliant diyuk from the gemara in Shabbos 14 that one is not allowed to do an action of shtus that makes it look as if it is part of a torah procedure], the marriage is completely binding. Yet in (E.H. 3:25) he writes of a reformed wedding where there is a double ring ceremony (even if the eidim would be kasher) the entire wedding is invalid becasue they are merely committing to want to be in a state of marriage but never doing the procedure of him being koneh her [R' Moshe explains that even according to R' Henkin that a secular marriage in a court will turn into a halachic marriage based on not wanting bi'as z'nus, that will only be where they recognize that they never had a religious ceremony. But where they think they are married based on a religious ceremony, the assumption is that all future bi'ah will be based on the premise that the ceremony under the chuppah created a status of marriage, so that the marriage will never be halachically binding].
R' Moshe (E.H. 4:32) was asked that these teshuvos seem to contradict one another whether a double ring marriage is binding but assur, or not binding at all. To this R' Moshe responds that he doesn't see any problem in reconciling these 2 teshuvos. In Siman 18 the ceremony was performed by an orthodox rabbi where the procedure was done properly, just that afterward he allowed her to do her mishagas by giving him a ring and saying some nonsense statement of him being mekudash to her. But in Siman 25 the reformed rabbi considered the exchange of rings to create the marriage - "הרי שפיר מוכיח שאין עושין קידושי התורה אלא שהוא נתינת טבעות בעלמא על שנעשו איש ואשה אבל לא היה שום מעשה על הקדושין". B'kitzur, it would be assur for him to give her a ring under the chuppah, but so long as it is clear that the kiddushin was achieved by him giving her the ring, it would not prevent the kiddushin from taking place. It would seem clear that if she wants to give him a ring later i.e. not under the chuppah, just for a symbol that they are married, that would be totally fine since it in no way is connected to the process of marriage.