I suggest that fabric creates a new mods folder for each game version. This way, you can install multiple versions of the game e.g. 1.13, 1.16, 1.20, 1.20.2, 1.20.4 and not have to move files around to launch the game. This will not only make updating easier because you can see all of your mods you need to download, but it will make going back a few versions easier, e.g. playing 1.20.4 but needing 1.20.1 for an outdated server, much easier.

This messes up not just mods, but configs, etc. Using a third-party launcher is probably the better idea. Or, you could use the system properties for the loader. Either way, not in scope for Fabric API.

This issue duplicates FabricMC/fabric-loader#181 as well.

Doesn't the Minecraft installer for Fabric Loader already do this anyway?
I'm pretty sure it creates a new profile (with a different game directory) in the vanilla launcher unless you tell it not to.

Doesn't the Minecraft installer for Fabric Loader already do this anyway? I'm pretty sure it creates a new profile (with a different game directory) in the vanilla launcher unless you tell it not to.

Profiles share .minecraft folders by default, so it would be different.