Confirmed owning multiple pets at a time, but can only take one with you.

Of course you can own multiple pets at once. They're not going to limit their micro-transaction cash store like that. Nothings more important than that net profit.

I'm still curious about the limit of their functionality.

Will they simply just work like chests where they are just static objects that you can pick up and drop down, where they will just stay put until you pick them up again? maybe press a button to interact where you can feed them like pigs in cages or pet/interact in some way with them? or will they actually have some form of AI where they can follow you and such?