Well, we have finally received our answers to Odin!
As it turns out, Odin's body is NOT the Primal, in the strictest sense. It is, in fact, his sword, Zantetsuken, that houses the spirit of Odin and manifests the Dark Divinity. When he is slain his body fades, but his sword remains and draws the aetheric energies required to manifest himself from the Shroud, not prayers or crystals. It is for that reason he takes so long to reappear--the aether of the environment is not nearly as concentrated as that within a cache of crystals.
After he is slain, those who come upon his sword are tempered as his "sundered" and are transformed into the Dark Divinity. I guess we can assume that, in the case of his FATE, the player who landed the killing blow is tempered by the sword, and when it has gathered the requisite energy to manifest Odin's body, the sword will call out to the player, which is why they become the new 'Odin'. This also explains why he tells the players upon his defeat that "forfeit is thy waking soul".
And this just reminded me: he's kind of like Shiva in a sense. He requires a host to manifest himself.
Unfortunately, it seems we will have to wait to find out why he is still so obsessed with finding Urth, despite having killed her centuries ago. I haven't talked to Urianger, yet, but maybe he'll provide some more info. After beating a Primal, he often has a bit more info to share with you about them, but that may be just for the EX Primals. Hmm... EX Odin? Make it happen SE, haha!
Needless to say, I loved the fight, and am really glad to get some concrete answers!
I'll edit my post later with some quotes from the quest, if someone hasn't already posted them by tonight.