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I think the best part of Pokemon is probably just the simple fantasy of having a fun adventure with your best animal friends. You just naturally form a bond with your favorites when they win a tough gym battle or evolve because it feels like you've helped each other grow. Even NPCs feel like they have a personal connection with their unique pokemon, to the point where a random fisherman with 6 magikarp is basically a tiny character story on it's own.
One of my favorite parts about Team Rocket as the original antagonists is that they aren't just in it to rule the world, their worst trait is that they abuse pokemon and violate the player's most basic premise: pokemon are companions and living creatures, not tools. I guess that's what pokemon means to me, it fulfils a fantasy of a world where the greatest thing you can be is best friends with a lv 100 rat.