Well, there isn't a national ID system, partly because the citizens don't want to be on the wrong end of when that API says "no". I'm not sure anywhere has such a fully available live system, rather than relying on people bringing documents in to the bank.

The live update would add an extra element of terror to the system, of course.

Edit: actually the UK system is pretty much this, except it's a token rather than an API, presumably to prevent you looking up random people without their consent: https://www.gov.uk/prove-right-to-work/get-a-share-code-onli...

Note that is for right to work, not right to reside, neither of which is the same thing as eligible for a bank account.