In the US, probably around 10,000. US Banks all operate by virtue of a banking license issued at the state or federal level. Banks must do as they are told by these centralized authorities or else they lose their license.
4,300 or so FDIC insured banks.
But um I'm not sure it's so different from Ethereum, where a mining node must implement the protocol or else be unable to participate.
But you do need permission, which is implicitly conferred by following the protocol. It's really quite similar to traditional banks, it's just that the protocol is digital/mathematical rather than legal.