I read currently a lot about monster catching/raising games. Next week is Pokemon day with new announcements, we get new Tem Tem content and there is a hype game we probably all have heared from. Monster farming, fight and hunting is not new for jrpg's. With Pokemon and Dragon Quest, there are two franchises with a long running history with this sub genre.
Here are some monster raising/fighting games i like:
World of Final Fantasy
I often want to write about World of Final Fantasy (Woff), because this game inspired me to name the community like that. There are probably people that doesn't like the artstyle of Woff, but i just laught the first time i saw it. And Woff want to be like that. This games is constantly joking about Final Fantasy Characters and creatures. The story often is funny, but also has a bit of deph, so i enjoyed playing this game even more than once. Maybe one ot the two Final Fantasy games i enjoyed the most in the recent years. But also the fighting system is a bit unique, so you are able to stack the monsters and main characters to strengthen them or make more dmg. It's look just funny and make fun at the same time.
Digimon Cyber Sleuth (Hackers Memory)
I think Digimon Cyber Sleuth is a bit like the favorite Pokemon, but it has a Story and not a perfectly blanace battle system. The Digimon franchise has sadly a lot inperfect games and people have different opinions what is the best games out of those, but a lot would agree, that the two Cyber Sleuth games are part of the better one. I think it kind of addicting to farm/collect and complete monsters in a lot of theses games and Digimon games are part of that. But i was never fan of the world games, that have a bigger focus on farming but more on the story games that have a plot you are able to follow. But at the same time you are able to get quite a lot of these monsters. The story is maybe not perfect, but i would say enjoyable because it has the same vibe as some of the digimon anime. I wouldn't say the game is perfect, story and gameplay has a few flaws, i probably enjoyed it that much because i got a bit frustrated from another monster game and this game gave me everything i wanted. A decent story and side quests in which this monster interacting with us. A lillymon that go inside the mobile phone of a person because it falls in love with that person, has quite some charm other monster games doesn't have.
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.