They could verify your identity if you've given it to them.

But it would also be completely reasonable to authenticate the customer exclusively using mechanisms other than government ID. You can already make a withdrawal using your bank card and PIN. If you lose your card you could sign into their website using your password and request a new one etc.

Consider what happens if you lose your government ID. Your bank has much better ways to authenticate you at that point than the government does. Government ID has a major bootstrap problem, whereas patronizing a service doesn't because a new account with no money in it belongs to whoever is signing up for it regardless of who they are, and you can at that point give them a bank card and have them provide a password and email address etc. that allows you to identify them in subsequent transactions without ever needing their name.