Autism

There's no requirement, and there's no reason a white person can't go to a black barbershop, but black people's hair is often much different to cut than most white people's hair, and preferred styles quite different.

It's not so much "this barber is for black people" and more so "this barber is owned by black people, so black people can assume they'll be given the experience they're looking for"

Because the hair is different. It's not necessary, but it's a common thing that barbers know how to cut hair similar to their own the best. I know first hand what happens if the barber isn't used to hair that is unusual.

I'll just say, apparently a military cut is hard to do for some people when the hair this thick

Black people have different hair from other races, many times non-black folks don't know how to accommodate these textures. Beyond that, historically, the barbershop in Black communities is a meeting place. People gather to discuss current events, politics, pop culture, and life in general. In this space, everyone is on the same footing. It's a cultural thing, so there is a distinction.