Can players pet Aniimo in the Homeland?

This is not available from the start. The Homeland unlocks when you complete the Home Invitation objective, and until then there is nowhere to keep an Aniimo of your own. Once it opens, the space is yours to build in: houses to construct and furnish, rooms to decorate, crops to cultivate, and room to house the Aniimo you have collected.

The ones you house there are not entries in a storage list. They occupy the space, which is why you can walk up to one and pet it. The same holds for creatures placed on the farm. Assigning an Aniimo to farm resources does not tuck it away behind a menu. It stays a creature standing in your space that you can approach and interact with directly.

Out in the world your hands-on contact with an Aniimo is twining, where you take direct control of its movement and use its mobility abilities. That is a tool for getting somewhere. Petting is not, and the Homeland is where it happens.

Worth knowing before you sink time into the place: Homeland development is paid for in Lumen Amber, the same resource that covers Pathfinder upgrades, creature progression, equipment, world activities, and upgrading the Branch in Idol. It advances regional and world progression directly. Every bit of it you put into the house and its furnishings is coming out of the pool you would otherwise spend on getting stronger, so a fully built-out Homeland has a real cost attached to it rather than being a free side activity.