Can someone describe what that style is? What are the characteristics? Why does this style exist? Does it lead/can be used as a platform to other things?
I think that's just a name Bartok used (in Mikrokosmos), it's not a widely accepted piano playing style. My guess is that he wrote in a way that sounded like Yugoslavian folk music, which is why he named it that. Note that there are also pieces in Mikrokosmos that are "in a Hungarian style", but that doesn't mean there is a Hungarian style of piano playing.