The French revolution overthrew everything traditional, not just monarchy. This includes Christianity, which at the time still maintained a traditional patriarchal order. They changed the week to a 10-day week because the 7-day week comes from the seven days of creation in Genesis. Naturally, the hierarchy of the sexes got questioned also, and if the peasants can rise up against the aristocrats, why can't women rise up against men?
The USA was created as an experiment in French revolutionary political thought, like separation of powers. Up to the 1850s, the movement to ban slavery involved a lot of women, and this gave women their start in political agitation. Americans would continue having an interest in French thought, and it was by this vector that feminism crossed the Atlantic. From the 1850s to 1910s was a straight shot of women's political movement after another, until they could vote and just outright ruled the country.
French revolutionary thought also formed the foundation of Marxism, which found its expression in the USSR, which gave women the right to vote in 1917. It spread this law by conquest and subversion of neighboring states.
I have no idea why the Scandinavians gave women the right to vote in 1906 and 1907.