It's... vague. I'm fairly certain this region does not have a monarch. At least, I've never heard anyone mention one. It seems the Elite Four are in charge of most things, but it is also possible that they are simply a liaison from whatever actual government exists here. They are, at the very least, in charge of just about everything pokemon league and battle-related, which admittedly, seems to be most of what everybody cares about in this world.

And, uhm, I guess the question of whether it is strict or not depends on what you are used to. Within these regions, we are free to go where we wish, we are free to learn and discuss what we want and to get whatever jobs we'd like. They have a police force, but I'm not sure if they have an actual army.

Some of my friends find this world a bit strict because they do come down hard on any physical fighting in public, even if it is consensual, but then, their world is more or less lawless.