I'm not sure if this is a character issue or a bug so I'll post it both places just to be safe. However, I'm unable to equip an Item Pouch on my fluffy lily pet. I've tried storing them in the CS, leaving, and then going back in to take them out to try again. Didn't work. I've tried relogging, didn't work. I've tried ordering the pets differently and it still didn't work. Video of me trying to equip the items on my pet.
If you experience further issues with equipping things on pets, manually put them on using the equipment tab (E). This is something I've been having to do from pickup accessories to label rings.