Chocobos ARE quintessentially Final Fantasy, and it would be dumb if SE just made mounts out of anything, unless like Nuru pointed out, maybe they have different strengths, traits, or abilities that make their purpose unique.

But to limit Final Fantasy to JUST chocobos? That's stifling creativity right there, which is what the series is based on. Just like FFXI goblins and mandragoras kinda became mascots for FF, there's no reason that SE can't come up with a new type of riding mascot on par with a chocobo or something.