They'll most likely add something like this, give it time.

FF11 had a feature almost exactly how you described. You would breed your chocobo with another player's, and once it was old enough you were able to ride it. It could come in black, blue, red, green, and yellow colors. It also had different stats such as speed and durability (how long you could ride it out on the field). You were supposed to spend time and play with it so it had a nice temperament, too.

There's definitely the potential for all that in this game.