The Pokemon League is a sporting organization with a religion-based heirachy - imagine the Champion has a position akin to the pope. What the Champions say will be reported in the media and their opinions hold significant weight in the public eye, but they have zero political power.
It just seems ridiculous that an organisation based on a single sport, and bursting at the seams with nepotism, could have any political sway.
Still, the League has power within its own sport - the Champion’s opinions decide how the sport is played, and the election of Gym Leaders, among other things. It has its own parliament, with Gym Leaders holding seats and voting on the Champion’s decisions; its own law enforcement, again, Gym Leaders patrolling their city; and is constantly at odds with the government in Kanto and its own police force (the Garda).
The Champion also holds political immunity, however, and cannot be prosecuted in any court of law.
The other regions do have slightly different Leagues.
In Sinnoh, the League is heavily related to the Sinnoh Monarchy. Princess Cynthia of Sinnoh became Champion at the age of 13… in no small part due to nepotism. Most members of the Elite Four there are members of the Royal Family, and the annual Pokemon League conference is held at the Crown’s seat of power in Lilly of the Valley Island.
Kanto, Johto, Hoenn, and other ‘lesser nations’, form the Sinnoh Commonwealth. Their champions report back to Sinnoh as the highest power in their Pokemon League.
In West Kanto, unofficially known as Johto, the League is shared with Kanto. Johto is not an independent region, and its government has limited political power and reports to Kanto.
Orre’s ongoing civil war has left its Pokemon League and political powers in a state of constant flux.