Does Aniimo have a story mode?

Progression is quest-based, so the game hands you objectives rather than leaving you to work out what to do with the creatures you catch. The main questline is the spine of that. Side quests sit around it, which gives you somewhere to go when you want to play without advancing the story.

The main story campaign carries voice acting in multiple languages. That tells you something about how the campaign is built: it is the part of the game with production behind it, not a thin frame stretched over the catching loop.

Story is not the only structured content. Operation Egg Heist runs separately, and it has a higher-difficulty Chaos mode for players who want the harder version of it. So finishing or pausing the questline does not leave you with nothing pointed to do.

Not settled: How Operation Egg Heist relates to the main questline, whether it unlocks through story progress or sits entirely apart from it, is not settled.