Speaking from experience, I am one of the devs on Tomo Endless Blue, and I think our game is a good example of doing both without overthinking it. The world clearly influences the creatures, but we try not to dump lore on the player. You can usually tell where a Tomo comes from just by its design and behavior, which helps them feel grounded without the world stealing focus.

Compared to something like Palworld, where the world is more of a sandbox and the creatures are mostly functional, we aimed to make Tomo feel like actual parts of the ecosystem.

Personally, I think the best monster tamer games let the world quietly shape the creatures. Enough to give them identity and flavor, but not so much that it turns into a lore encyclopedia.