What's up with Bahamut?

[Beddict]

Let's see... since I am in love with lore info gained from battle content... (though Enkidoh already did an exemplary answer).

Midgardsormr's Simulacrum as Recorded by the Omega Weapon during fight on the Dragonstar:

'Tis the fate of all dragons to shed one form for another. Even in battle mine assaults are ever changing!

That fact, combined with the fact that summoning/creation magic is inherently difficult, and then further combined with the fact that clearly Dragonkind was new to it... yeah, it's not hard to see why Primal Bahamut looks different from actual Bahamut in FFXIV. I say that dragonkind is new to it, because at least the First Brood were. If dragons had creation magicks, then Midgardsormr did not teach them to his children.

Also worth looking at Ehll Tou, the dragon involved in the Ishgard Custom Deliveries, since she helps further prove your point by backing up what Midgardsormr's Simulacrum says. Near the end of the questline she undergoes a transformation that allows her to better old crafting and gathering tools, and the other dragons involved the quest point out how different she is. From An Ode to Unity:

Ehll Tou:
Ah, I knew you would recognize me! Still, an explanation may be in order...
You see, a dragon's growth is dictated by forces both internal and external. Sometimes this growth can be quite...dramatic, as you can plainly see.
Truth be told, I had been feeling out of sorts lately, but now I see that that discomfort was merely the new me trying to break free. And now here she is─a dragon reborn!

Hautdilong:
Oh, I'm so glad you're all right. More than all right, by the looks of things!

Ehll Tou:
This new tool is wonderful! It's as if Master Arvide knew I was on the brink of incredible change!
Had I received this hammer but a day earlier, I would have held it clumsily in my tiny claws. Now, however, it feels like an extension of my very body. My mind is brimming with the possibilities of what I might craft next!

Dragonet:
Your growth is unlike aught I have seen before! Such a spectacular transformation has already made you the topic of much discussion among dragonkind, and I imagine not a small amount of jealousy.

Dragons are capable of a hell of a lot of adaptation as they grow, and there are many types of dragons we come across in game. Just look at Marcelloix in The Firmament, he's a dragon that stands upright and doesn't really have any wings which is way different from the First Brood or the generic Western type of dragon a lot of people are familiar with. I'm not surprised that a large group of Meracydians of all different shapes and sizes involved in the summoning ritual spat out a Bahamut Primal that looks vastly different than First Brood Bahamut.