I had an older brother and there were clearly double standards (pretty close to the same scenario as OP presented), but now listening to my parents talk about the next generation of kids (and they're raising one), it's worse than I thought. The one that grates me the most is that the boys can be cocky little shits and the adults will just laugh about it because they think it's cute. The girls can just be assertive and they're labeled as "stuck up."

Though, how I was raised with my brother is similar to OP's fiancee and her brother. My parents tried to give my brother one chore of doing the dishes every other week, but he failed to even try, so it was passed on to me and I got my ass handed to me if the kitchen wasn't spotless. I can't say that was definitely sexism, but reading those paragraphs felt familiar.