PayPal offers you a virtual account that you can pay money into. You can use that money to make purchases online, send and receive money from friends or other businesses. In effect, it acts like a bank account. However, it's not an actual bank account. In Europe, any money you put into that account is also not ensured by the government, like a normal account would.
If I pay with a credit card, there are processes in place to deal with fraud and charge backs. PayPal is well known to automatically close accounts with little recourse to access the money on those accounts.
We might be talking at cross purposes; I'm not sure! How is it like a bank?