I don't understand why the "Companion" menu is called "Companion" instead of "Chocobo" since the chocobo is the only "Companion" in the game. I'll suggest some other possible companions:

Moogles: Besides the chocobos, moogles are the most popular creatures of Final Fantasy. I think they deserve to be companions. The Mooglesguard showed us that Moogles can also play war/magic disciplines. We could choose a clan (Gridania or Dravania, each one with some differences) and a discipline for them. As for the customization, they could change their colour like chocobos, and their hats. To unlock ranks, we could use Kupo Nuts which works like the Chocobo's Thavnairian onions.

Gnath: Since they have built an Aventurer's guild, this is my second option. A gnath from that Adventurer's guild. Customization? Well, this could be complicated. And to unlock ranks...

The Doman kids: The Doman kids from Mor Dhona. They want to be adventurers, and that's why they've created their own "Adventurer's guild". We can create one of those kids like we create a Retainer, and take them as our apprentices (companions). To unlock ranks, they could have their own quests where we help them to unlock the next rank. This last option could give the game a new sidequest line based on the training of the Doman kids.