Probably has its roots from way back in the day so that women couldnt effectively run away from the men and get very far.

Can't speak to Muslim culture, but European culture way back in the day didn't want women riding horses because of sex.

There are a lot of branches on that tree, but the biggest one is that since horseback was believed to be capable of rupturing the hymen (hymen science has progressed quite a bit since I last looked into it, so I don't know if that's actually a thing), it was the same thing as having sex for women. They believed that women got sexual pleasure from it (which, I guess, was a bad thing), that they'd start craving horses as lovers instead of humans, and all sorts of weird shit that only twisted, perpetually horny dudes would think of.

So the sidesaddle was invented. It allowed women to ride horses while, literally and figuratively, keeping their legs closed.

Unfortunately, riding sidesaddle is a massive pain in the ass, so that fad didn't last long. Maybe about fifty years or so of general popularity (because, obviously, you can still get a sidesaddle and learn to ride in it today, if you want, for whatever reason) over the course of all horse-domestication history.

Of course, like so many things from European history, this primarily applied to rich/noble people. The poor didn't have the luxury of giving a fuck about most of it.

I gave a Google. It's seen as a taboo thing in many Middle Eastern societies based on a verse from Muhammad which curses the private parts of women who touch saddles. Curiously, this line has been debunked and the Quran literally mentions nothing about women riding horses.