I wonder however how they would react to somebody becoming a different species. Like say some guy in the 24th century says he wants to become an Andorian, would they allow it? Would there be a stigma associated with a desire like that? What would the Andorians say about that?
Interesting thought. Given how easy that kind of cosmetic surgery is in the 24th century, and given how similar so many humanoid species are to each other, I'd imagine that sort of thing might happen all the time.
We've seen a couple of cases in the novels where spies were surgically altered as deep-cover moles in alien societies and ended up going native, e.g. in The Romulan Way by Diane Duane and Brinksmanship by Una McCormack. But I don't think we've ever seen it done openly as a lifestyle choice.