Its a shame the way they did it, its really not that usable. The only real reason you'd ever mount up in battle, other than the looks, is to get a low jump/low move character to a specific spot very quickly.

When a character mounts a chocobo, their speed and attack power is the same, but the move/jump ratings of that character become what the chocobos was (since he is now riding the chocobo), and your chocobo no longer gets any turns (till the rider dismounts, which I believe theres an option for once you are riding)

The idea I guess is that it can help you move a slightly immobile character far away, very quickly. But the chocobos that can really move great distances can either a) fly up close to enemies and do great dmg,or b) drop a big meteor on a very far away enemy for LOTS of dmg, or c) provide excellent healing to your allies by not mounting it. Had they provided other benefits to mounting a chocobo in battle, i.e. maybe adding a portion or all of the chocobos attacking power to the rider (i.e. rider has 10 PA, chocobo has 15 PA, rider now has 25 PA), then it might have been more used