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Digimon (specifically World 2, DS, Dawn/Dusk, Cyber Sleuth come to mind with monster catching) and Monster Rancher series come to mind. Also liked Dragon Quest Monsters: Joker.
I do like any game that makes me feel like a beastmaster. Even simple shit like Hunter in WoW satisfies that urge of being a commander of monsters and befriending them while also turning them into powerhouses. Dragonseeds and Azure Dreams for the PS1 come to mind and have their own little twist.
I honestly can't find enough games that let me do stuff like that.