Actually, a lot of this was already explained and confirmed by the lore panel. I learned a lot about it in a previous thread about Odin. For one, Odin is a special case, but not in the sense that Eikons are special from Primals.
It has been all but confirmed that the Odin battles we've been fighting so far are actually NOT Odin. Though it has never been explicitly stated in-game, The Dark Divinity quest line confirms that Odin's aether does not originate from the body, but from the sword.
The lore panel specified that Zantetsuken is that it is from the North (was likely a Raen weapon) and is "cursed." Odin won the sword in combat, presumably killing its original wielder, and took it on his quest to rescue Urth. He uses it to great effect, but he eventually succumbs to the corrosive influence of the blade.
From this, a few conclusions have been drawn. The most commonly held assumption right now is that Zantetsuken is a Primal. The blade works in a similar fashion to the way Iceheart became a host for Shiva. It possesses it occupant and shapes them into Odin, the only warrior it has deemed worthy of it. Any non-echo user would die in such a process, a living sacrifice to the sword. Odin's persona remains, but it is decidedly not Odin. Also, since each battle ends with the body being defeated, and not the sword, the aether never dissipates. No one has actually ever won the fight, because we haven't been fighting the real Primal.
There's also a theory that the sword feeds off of the endless supply of prayers fed to it by generations upon generations of warriors seeking to be the best that they can possibly be (that would be several thousand years’ worth of "prayers" offered unknowingly to this supreme weapon), and a few theories as to what happened to the real Odin, and whether or not he is actually alive. Some of these theories take us back to the Warring Triad (as there are similarities between Urth, Wiyu and Odin and the Fiend, Demon, and Goddess) and SE's actually makes a direct reference to Odin and the Warring Triad in their lore panel (though the reference is a little ambiguously worded) so people are just waiting for the details, but what we do know right now is that Zantetsuken is (most likely) the primal in the Odin fights, and that the body we are seeing is definitely not him. Further, the incarnation of Odin that it summons uses a method that we are very familiar with, as it has been used by several other Primals (Ifrit, Garuda, and Titan Ex, Shiva, Phoenix, and the knights of the round, all use the same or a very similar method).