If you take a look at the major differences between culture in the 24th century and our current society, there is one area where the 24th century is more advanced than ours, and that’s gender equality.
After the dust settled, there was a strong consensus regarding abandoning cultural mores that exist today. One of these was patriarchy. Initially, equality was born out of necessity: with the collapse of organized society and any form of government authority, people had to fend for themselves. There simply wasn’t room for any ideas about what women are supposed to do and what men are supposed to do. After that first initial stage, the thinkers and polymaths of the era embraced the idea and practice of equality as a way to move forward. Make no mistake, despite people in the 24th century knowing that life was much easier in the 21st century, they don’t look back at it as a golden age. After all, our way of life could have survived the comet, but instead we killed it off with nuclear weapons.
If there’s one thing that’s universal in Northern Europe, it’s that men and women are equal. That’s really all there is to it. The most famous military unit of the Kingdom of Peimar is the Night Witches, which is a dragoon cavalry squadron with only women in the ranks. No one thinks this is strange in any way. Only competence matters. However, having both men and women in all lines of work and in all possible offices in government is just a superficial effect of a deeper cultural change.
For instance, domestic abuse is considered to be more heinous than today. Someone who talks about hitting his or her spouse during an argument is probably going to end up a pariah and losing most friends. It’s simply not done. A man who insults a woman because she is a woman will probably end up shunned, too.
There’s no stigma to give birth out of wedlock, either, and it’s not uncommon for a man to take care of a child in this situation, if the mother is inconvenienced. In general, you’ll find both stay-at-home fathers and mothers in the 24th century, since it is assumed that whoever is more capable of providing for the family, does that.