I wonder if this is primarily an American thing, or if this phenomenon exists in other countries.

Seems like one could solve this issue by making university cheap or free.

Not only American, but not common either. The best analogy to American business schools will be party (communist party) schools in Soviet Union and China.

I can't pick words to express just how similar are these two systems.

And in all of those countries, you have huge masses of clueless parents seeing these institutes as something that magically, effortlessly make their children "big men" of higher socioeconomic standing overnight once they graduate.

As for me, my own experience was that it takes a lot to convince parents that Western "big name business school" is a gigantic waste of money if you go there for actual skills and knowledge. And the argument "you go to business school to make connection with kids of other rich people" does not hold water either. All of such students are well connected people to begin with, and are very class conscious themselves. If they see no use of socialising with you, what will make them want doing so?

I wrote another post below explaining how it was for me.

You're wrong. By getting into the school, and then acting the part well, you can make a friend who gives you an opportunity to get into the class.

An MBA in the U.S. honestly doesn't even feel like "university" in the traditional sense.

It is essentially an exchange where the student pays a massive amount of tuition money and the university gives them two years to network with people. I believe the goal is to make friends with the folks who were going to get that finance job through their connections anyway, and then maybe they can get YOU a finance job too.

It likely exists anywhere universities rely on alumni’s donations, i.e. anywhere universities are private businesses, like in UK. Probably not a big deal in places where unis are state-funded and places are not limited.