I think the last time I uses a gender specific hairdresser was when I was in Intermediate School.
I don't know whether my hairdresser is unisex or not. I know they aren't female only. I doubt they are male only having female owners.
The thing I most like about barbers is there is less of a chemical smell than in a proper hairdressers, the atmosphere is more relaxed/quiet, I donm't have to book and it's cheaper. I gave up on my hair a long time ago, it's all about staying tidy for minimal bother and outlay.
They do a nice beard groom as well (more tidily than I can do myself).