It isn't because they are barbers. That is secondary. These are men's groups. This is men helping other men. Thanks to continued differences re hairstyles barber shops are one of the very few all-male places left in modern society (other than locker rooms which have a distinct competitive air). So it is no surprise that many urban men feel at home and are more willing to talk at the barber than anywhere else.
Barber shops are also full of the most toxic ignorant drivel you can possibly find in the physical world. This article and the positive comments highlights a need for actual mental health advocates aimed at men.
Eh, as a Black male, the possibility of ignorant shit tends to accord with how young the age average of the barbers is, older/elderly barbers aren't treated as peers usually and so have a modicum of authority to curtail discussions when they feel its out-of-line or inappropriate, not to mention they're usually more thoroughly religiously observant. The church use to be a space for discussion with an authority figure like that but most Black men, particularly younger ones, have evacuated the church.