Those soccer leagues absolutely operate with a monopoly in individual countries. They even have limits on the number of foreign-born players they can have on their rosters -- which is something you'd never see here in the U.S.
Some sports (like soccer) are more complicated than others because they operate in multiple leagues around the world and are subject to the laws of different countries. In cases like that, the individual leagues end up signing agreements with each other so their player contracts are recognized in every league and the players can't just move from one team to another in the middle of a season.
The US has multiple professional soccer leagues going on currently right now that aren’t affiliated with one another. Most of the other sports had the same thing. The only thing that happened post merger was ticket prices went up, parking went up and cable costs skyrocketed (ie the anti-trust should have held, all things being equal - but they weren’t b/c they weren’t political animals at all).