Nickname or full name?

I would probably not put the name on it at all. Unless they typically wear things with their names on them, if you want them to use them outside of the wedding day, are they likely to go out with a clutch with their name on it?

Agreed, are these meant for your wedding?

What do they prefer? Is there any way you can ask them without completely giving away what it is like "If I were to give you something personalized, would you like your full name or nickname?" and not say what it is?

I think it is just up to their preference, but I will add to what others say about them, but what do they look like? I'm trying to imagine them because my mind jumps to those cheap mass produced bags that are screen printed and stuff that are really only there for novelty and maybe a few cute photos and not really something practical. I guess it would be different if they were embroidered and the names weren't front and center.