I respect gender identities, I'm for trans rights, etc. but in a real situation, meeting someone like this and being told they don't like being called a man, get me out of this minefield now.
It’s not a minefield.

It’s no different than getting someone’s name wrong and having them say “that’s not right, my name is John”. It’s not a gotcha.

If someone says “don’t call me a man” there are literally an undefinably large number of ways to just not use “man” (or guy, or dude, or bro). Nobody is trying to catch you out or looking for a reason to correct you. It’s just a basic thing to refer to someone in the way they prefer. Like how some Women for **decades** have preferred maritally ambiguous titles like Mz instead of Mrs or Ms.

Instead of “you’re a handsome man” you could just say “you’re beautiful” and everything would be fine.