No. The EXP is a set amount per level, and does not scale according to the level you summon it at. The value is not available (from datamining), invisible (to you), and the mechanic by which it inherits EXP is unknown at this point. The EXP required to progress to the next level increases at an unknown rate, but it most definitely increases.

Do not expect your chocobo to level if you are using it during the leveling process of alternate classes. It does not receive quest EXP, levequest EXP, or FATE EXP. It can not be brought into dungeons, and summoning it while duty finding will eject you.

Full disclosure, conjecture: users have reported that you get less EXP per kill with your chocobo out. I personally have not found the EXP gained to scale from chaining mobs your own level. It is an unknown percentage, but I believe it to be about 10% shared, not 10% bonus. You'll have to look very carefully, because the EXP your chocobo earns from a kill is shown underneath of the EXP that you earn per kill. To verify, kill a mob without the chocobo, check the value, and then summon it with a gyshal, and look for two EXP numbers.

At level 10, you will have exactly enough skill points on your chocobo to fully max out one of the trees, and there are no plans to provide a respec of any sort. If you place points in two trees at level 1, you are already locked out of the rank 10 skill. Why they gave you two points at level 2, and two 1-point options is beyond me.

If it makes you feel better, this is what I thought when I found out: so wow brutal wow such balance wow wow so many brood war was balance such wow such suck square enix pls wow wow also no mounting during combat such balance wow wow wow very terror