If you attack the DMV in a country that has working eID it will only affect the DMV. Maybe you can't sign up for a driving exam or order new plates. That's it. It won't affect the police in cases that don't concern the DMV (like insurance) and vice versa. Sure if you attack everything at once everything will be affected, that's always an option, eID or not.
Fundamentally a digital ID does not mean a single "basket", just that wherever those "eggs" are that you know it's one single "chicken". That "chicken" can also have multiple ways of identifying themselves (including offline methods). That's how it is in countries that have a working eID implemented.