Technically, no Barber in-game lets you change skin color, only fur color. Though only three of the four furred races can do this, Trolls being the exception, despite it being said in Vanilla lore that they get their color from their fur, and Blizzard Historians in 2014 confirming it is still the case they have fur.. "Skin Color" being a customization option for Trolls in WoW sure is a long time running bug.
I'm personally indifferent to changing Skin Color being added, I can get behind the argument that the barber already allows you to change your character's face. I don't think it'll happen though as that takes an option away from paid character re-customization.