Monster Rancher
I'll toss in another vote for Monster Rancher ;) A Pet Raising and Breeding simulator that got it's roots from an earlier 1996 game "Gallop Racer" which was a Horse racing and breeding sim. Monster Rancher replaces Horses with Fantasy monsters, and swaps out Races for Real Time battles.
While the Raising aspect is highly menu driven as you make constant adjustments to your strategies for raising stats, learning new attacks, or doing "side-quests" in the game to unlock even more monsters, the Battle is in real time where you control your monster moving between near and far ranges to use attacks that are unique to those ranges. The attacks themselves also have 2 primary types (Power or Intellect) and 5 sub-types (Heavy, Crit, Wither, Hit, and Special). Using attacks has an associated resource cost (Guts) and your monster's guts regeneration differs based on it's breed (or mixed breed).
Loads of variety of strong looking, silly, or fun monsters to pick and choose from.
- in 2021 the original first two games games were re-released as remasters on Switch, Steam, and iOS as Monster Rancher 1 & 2 DX. with upscaled graphics, some overhaul in the UI, and a lot of QoL features added in, but for the most part still looks and feels like the PSX game, just a lot smoother and zero load times.
- in 2022 a new game exclusive to Switch released asa crossover with Ultraman where you raise Kaiju from that franchise but with Monster Rancher rules.
Monster Rancher is alive and kicking!