Yes I always give my married sims a ring. For women I usually give them the one with the big diamond (the one that the proposal animation uses) and for me I give them the plain band that goes on the ring finger.
Maybe it's trait based. The groom's traits are bro, playful and erratic and the bride's traits are neat, clumsy and foodie. It's weird though because the bride was the one who did it and she has the neat trait while he has the playful trait
Here's the link to the picture (I hope it works I've never tried to share pictures on here before) and I know it can't be a mod because I don't have any in my game