hmmm. For my current roster I think Kuai would ride a giant blue kingfisher. (maybe in fantasy a little icy since he's ice themed.) Kabal would ride some sort of terrifying bird of prey that has teeth for some reason. And Genji would ride a Quetzel
I can see Ace riding a bearded vulture, Prompto still wants his chocobo but maybe he'd ride a canary if it's our world birds giant only, The Janitor would ride a Shoebill, and Alucard would ride a raven.
Ram would get a bluebird, but upgrade to bluejay after Legacy. Roy would somehow wind up with a barn owl, but he's mostly awake at night so that's fine. >D Eckert teaches his umbrella cockatoo all his favorite dances. Wulf gets a harpy eagle, Gorn a magpie, and I'll think of others later.