What do carried items do in Aniimo?

Each Aniimo equips one carried item. That single slot is the main lever you have for making a creature stronger, which turns it into a decision about what that Aniimo is for rather than a matter of stacking whatever you happen to own.

The bonuses land on the stats every Aniimo has: HP, Attack, Physical Defense, Regen, Magic Defense and Break. Not all of them are flat increases. The Ambusher Cloak raises damage dealt when you attack an enemy from behind, so its value depends on how the fight is actually going and whether you are willing to work for position.

How they sit next to talents and battle items

Carried items work on their own once equipped. Talents are the other half of a build, and you get two equipped talent slots per skill set, so that is another choosing problem rather than a collecting one. Unlock talents early; combat gets easier once they are in.

Separate from both, there are items you can use during a battle to power up an Aniimo while the fight is running. Those are spent in the moment. A carried item keeps paying out for as long as the creature holds it.

The bonus belongs to the creature, not the Pathfinder

Your Pathfinder fights on the field directly, with attacks and gadgets, and is protected by a hit-based shield instead of a health bar. Extra HP from a carried item does nothing for that. It goes to the Aniimo, and in human form you have a skill that restores a targeted Aniimo's HP, so the pool a carried item raises is the same pool you are topping up.

This holds in Command mode too, where you stay out and issue abilities while the Aniimo fight automatically. The gear is doing its work whether or not you are steering the creature yourself.

Not settled: Which carried items exist beyond the Ambusher Cloak is not settled, and the full range of effects available in a single slot has not been pinned down.