In the story I'm writing there's a plot device that is a bank account and I'd like to know what kind of information would be needed for such. As in, if a character had an account and wants to give it access to another character, what kind of information would that be?
What kind of access do you mean? You can give someone a power of attorney. You can add the to the account as one of the owners of the account. You can give them your debit card and pin. You can give them access to an app for the account or that makes payments drawing from the account.
Depends on the bank and country, but you might have multiple factor authorizations. Some even with physical authenticators. You could also have character A open a sub-ledger for character B.