Just a comment from another Swede:
It is not mandatory with an ID card, not even to vote, see: http://www.val.se/det_svenska_...
Google translation of the second paragraph (yeah, I'm lazy):
Do I have to show identification when I vote?
You must prove your identity to vote. You can prove your identity in three ways:
1. Be known by the voice receiver
2. Show ID document
3. Any other person can vouch for your identity. Then the one who vouches show his ID. The social security number of the person who vouches written down in the electoral roll.