I am fairly certain that the XP Bar that is displayed is for the TOTAL Xp, and it doesn't necessarily mark out for each individual (similar to the FC's Allegiance to a GC)

Now I can not say for certain that this is true as I don't use my chocobo in combat really as I haven't leveled any alternate classes yet (aside the job required and I did that before you could battle ready your chocobo).

Like was already stated, chocobos only recieve XP from individual kills and not from FATE Completions/Quests/DoLand/DoHand. The amount of XP from 0 to 1 isn't that bad, from 1 to 2 is not fantastic (ran FATEs in Costa Del Sol region with 4 other FC folks using Chocobos, back before FATE grinding had become so intensely popular for a few hours I got 3 levels on my GLD and only the 1 level on the chocobo)