Here is my unbaised view on jumping.

It is not neccisary but, it does add a new dimension (Litterally) to the game.
Now, as for mounts. . . If jump is implemented. . . Chocobos (and maybe wolves) should be able to jump. I just can't see a Gobbue jumping at all. Arihams should fly and Drakes. . . I am not sure.

These things should all be situational though. (Ofc, Arihams should be able to fly anyhow as a regulars way of transportation.)