I'm inclined to believe that Enkidu isn't a summon at all and has been Eorzea all this time (well, about as long as Gilgamesh has been). A 'giant bird-like creature' or something of that nature had been mentioned once or twice in 2.2 (and possibly later, but I can't recall). Also, weren't the elemental crystals supposedly used in Enkidu's summoning fire (I don't remember)? If so, then... doesn't that kinda throw the entire "can only be summoned by the crystal representing their element" thing out of the window? Enkidu is quite clearly of the wind element.