It seems like the chocobo you raise is going to be the pet you make an attachment to. It's been said you can raise them to be damage dealers, tanks or healers and that they're supposed to have a gambit system.

As far as a class dedicated to pets, seems like Arcanist is going to be it for now with Carbuncle. Who knows what they have in store for the future, but Yoshi-P has already shown interest in Samurai and Ninja so it may be a while before we see another class like BST or PUP.