Initially, Bart didn't like Wally and was very leary of letting himself be connected with him. They were antagonists pretty much all the time they were together. After working with Wally for a while and seeing him in action, though, my impression was that Bart had gained some measure of respect for him and seen that he'd made the Flash title his own, even though he bore the same name as Jay and Barry and wore a costume almost identical to Barry's. That realization, plus the desire to reinvent himself, made it seem natural for him to accept his place as the new Kid Flash. Rather than being a concession to the Flash legacy just because that's how it was done before, I saw his taking the Kid Flash identity as an acknowledgement that he knows he's not Wally and he's confident that he can be his own man, even though the world knows there was a Kid Flash before, just like Wally's managed to be his own man even though the world knows there were two other Flashes.