Early in ARR's development it looked like SE indeed were planning on having each Job get their own special mount, but apart from CNJ/WHM getting a unicorn via an easy quest after getting CNJ to 30 and completing all the CNJ class quests, that never eventuated (the tank achievement mounts came much later). Alas I think that airship has long since sailed.

Instead, just set up certain existing mounts based on their theme to each Job (it's something I've done myself).