PayPal EU is a bank registered in Luxemburg and their EU operations are regulated as one: http://www.e-comlaw.com/e-finance-and-payments-law-and-polic...

PayPal issues credit and loans and credit cards they are an issuing bank and they also serve as an acquiring bank for processing payments.

Their account balance may not count as deposit and as such that part might not be under banking regulations in the US but in the EU they are regulated as one.