Where does Aniimo take place?

You arrive there as a Pathfinder. That is the role the game hands you in Idyll, and it is the word used for players rather than trainer or tamer, so expect to see it in menus and mission text.

Aedil is not presented as one undivided map. At least one named region sits inside it, the Lost Islands, which is where Egg Heist runs. If you are trying to work out where a mode takes place, the region name is the useful unit, not the continent.

Why the world's name keeps changing

The name of the world is not settled. It has turned up as Aetheria, as Aida, as Aether and as Idyll, and the spelling Idle appears as well. This is pre-launch material from the closed beta, and the setting's proper noun is one of the things still moving between builds.

The practical effect is that two pieces of information about the same place can look like they describe different games. They usually do not. Treat the world name as unfixed for now and the continent, Aedil, as the steadier label.

Not settled: The world may not ship as Idyll, since the setting has carried several names across builds. How much of Aedil is playable, and how the Lost Islands sit within it, is not established.