I haven't heard of that before, mind you I've only really rented in small apartment buildings. I can understand if they want to charge $20-$30 for the registration process or what have you, though seems like something most would cover anyway to just lock down a new tenant. $350 seems like a grab, maybe they do that in more in-demand buildings because they know they can.
Good to inquire about especially if negotiating any terms for sure!