I may have it all wrong, but as I understood it when I visited in 1968, Iceland was similar, with the exception that very few women were married then. The women and children were taken care of completely, but the men were sort of left out to dry. One of the things that struck me was that the guys didn't pick up the girls! It was reversed! The women had the apartments and the men, if they couldn't afford otherwise, lived in "dormitories" for want of a better word. The women would take men home for days or weeks at a time then kick them out and replace them. One of the women with a kid that I met said that any man who saw a child misbehaving in public, however, was expected to come over and take care of the problem. Imagine that in your local Walmart these days! They'd drag the guy off in handcuffs, if he survived the beating from the other women in the store. It was interesting, to say the least! Children took/take the mother's name, pretty sure.
If this is TOTALLY out of line with reality, circa 1968 Reykjavik, I'm certainly ready to be disabused of what I heard and saw there, but I encountered nothing at the time that would contradict it. I wonder how things have changed in 44 years.