Earlier quoted context omitted.
You need an ID to drive, buy alcohol, buy cigarettes, board a plane. And people manage to do all those things.
You can do all that with a driving license, but a driving license is not a proof of citizenship and thus of your eligibility to vote. In Europe, you can't get a regular ID without being a citizen. So having an ID automatically means you're eligible to vote.
Just because a driver's license isn't proof of citizenship doesn't mean that it isn't sufficient, since it will have some identifier connected to the identity of the voter. You would show the driver's license, then the registrars would find your name on the list, strike it out, give your voting envelopes, and then you go wait for a private booth to become free and then you go into that booth and put your votes in those voting envelopes.
A Swedish driver's license doesn't mean Swedish citizenship. It can still be used to check somebody's identity. Citizenship is checked by seeing whether the person is on the voter rolls, not by their identity card.