My old school favorite, not counting Pokémon, is Monster Rancher. I always liked how raising the monsters worked in those games and battling was its own thing that you did to show off the fruits of your training.
A modern game that I loved was Monster Sanctuary. The battle mechanics are very satisfying. The battles are 3v3 and move quickly. The mechanics make for multiple viable strategies. Every monster has 2-4 skill trees and 3 forms, meaning there is a very high level of customization too. On top of that, it’s a metroidvania, so you have a map to explore and areas you can’t access until you find a monster with the map action to get you through
Jesus, no it doesn’t. These media outlets want nothing more than to put the word “Pokémon” in their headlines to drive clicks. It’s as little like Pokémon as the game has always been