PayPal is not a bank, although people seem to think it is, and they seem to be more than happy to let people think that. It's long overdue for some serious regulatory oversight.
PayPal is a fully licensed and regulated bank in some regions including all of the EU, and is licensed and regulated by each US state it operates in.
Many of the startup alternatives people recommend seem to be operating on a "try to grow fast enough that we can afford to operate legally when states start suing us" basis, as they aren't similarly licensed.
In Europe, paypal registered as an actual bank (in luxembourg). Incidentally, i haven't had problems with them as a merchant through the years other than the fact that they ask for too much documentation in order to accept payments more than a few thousand, and don't have enough withdrawal options sometimes. If i used a bank instead i m sure it would incur much higher fees and bureaucracy.
Because federal regulators said so over 10 years ago. They don't accept deposits as defined by FDIC, engage in fractional reserve banking, or otherwise meet the definitions of a bank in the US. They're more like an escrow service, holding funds on your behalf in accounts at actual banks. They are a money transmitting entity, which means obtaining 54 state/territory licenses to operate in the entire US, and they hold all of those.