As others have said including in the OP, it's pretty obvious the idea that the Pokémon League is the Government™ would be like FIFA being the government and that simply makes negative six sense. Besides, the anime at least has mentioned town mayors among other things, and concepts such as prison exist, so there certainly exist layers and levels of government somewhat similar to our own.
, but since politics are pretty boring for the target audience and irrelevant to the canon storylines, they understandably just don't go into it.
I think this is the basics of it. Pokémon training is an activity that functions much like a sports or a hobby with similar interoperations going on How often when you go see a tennis match at the stadium or go to a museum excursion in school do you get to meet a councilman, let alone the Mayor or the Governor, and see what are they doing?
Politics is there, subyacent to much of what happens in the Pokémon world but still
subyacent. What else is going to regulate the whole idea that eg.: Trainers just wander around entering into people's houses?
What kind of world can allow children to run around with wild, dangerous creatures unsupervised?
Why the best kind of government of course! This is a world where children go out to explore the world at the age of ten on what appears to be government dime, tutored by their peers who have attained empirical experience, and doing the whole drifter trope smiting local evil and taking names while at it, and with how much social activity and economical activity would Pokémon training as a sport push in such a world, it's not difficult to see why.
Government-wise for my setting I have them just be somewhat similar to our own. There's Presidents and Emperors and God-Emperors and then there's everything going down the ladder to the little Councilman who just wants to make sure the town is not running out of water and stuff. Regions tend to function closer to what would be states in a Federal system, subordinate to a larger "National" concept: for example, I have it that Kanto, Johto, Rota and at least one fanon region are part of an unity that would be similar to "Poké-Japan", whereas regions like Unova, Orre and cross-fanon ones I work it like Angela and Domino would be part of a different unit closer in concept to "Poké-United States"; in some other cases regions would be large or independent enough that they would comprise their own little countries like ya know, lovable Lichtenstein or awesome and never forgotten Denmark as would be the case for eg.: Kalos and Suocé, my main fanon region.
Pokémon being such a transversal aspect of the world, Pokémon training is regulated by mostly orthogonal world organizations at the level of what would be UNESCO for our world, except with better jurisdictional reach and even then there's also stuff like the Pokémon Rangers. Such organizations would have regional off-shots in some cases but would still largely operate by promoting a common Standard for training practice. Legal issues that crop up when a random youth is journeying across a region are probably addressed by a board composed of local government officers and representatives of a Pokémon Training union or fiscalization body, and for all but th weirdest or more harmful cases they never reach an actual court or a police file.
All this however deals only with human government. Pokémon being sapient creatures organized at various societal layers and some of them even having urbanizations in their own way, there'd also be a lot to talk about what a "government" for a Pokémon colony or a more macro ecosystem would be like, and in my fanon some Pokémon do organize themselves in province-like structures ruled by their own and working with a bare minimum set of laws and which are commonly called Sovereignties.