And I wouldn't. See Shale's reasoning. A non-elemental attack can be assigned an element in the defender's system if it is reasonable to do so. However, it must be assigned this element for positive properties it DOES have (e.g. Flare is considered a magical attack in FFX, so we give it the Magic element in BoF4.).
You have to make an assumption each and every time you do this. It's doable, but again, more assumption=less likely to be true. It's simpler to just assume that all attacks are comprised of at least one element. This also is truer to the definition of the term "element" and does not result in the absurdity of an attack having no element at all.
So wait, Flare has an element IN ADDITION to magic? So not only have you arbitrarily decided everything needs at least one element type, now everything needs at least TWO?
Learn to read everything carefully...seriously. I didn't say this, imply this, or suggest this. In fact, I explicitly stated that Flare just needs to have *one* element and everything is A-OK. You can call it Magic if you want (which would make, say, Firaga Magic+Fire), but you aren't necessarily pigeonholed into doing so.
If Flare does indeed need one minimum element, then it already has it: magic. An element we already know isn't Almighty, since Bufudyne is also clearly "Magic" (I assume you'd treat it as such in BoF4) and isn't Almighty.
More conflation. Almighty isn't an attack, Bufudyne is; you can't compare the two directly. If you want to compare directly, look to an Almighty attack.
Going by your logic: Bufudyne = Magic + Ice; Megido = Almighty + Magic
Now, from here you *could* state that because Jenna takes full damage from Ice, but not from Megido, Jenna is neutral to Ice and Magic and resists Almighty damage (meaning Flare would deal full damage). This is a valid view, as I've said before, assuming you consider Flare to be Magic as opposed "Undefined" elemental.
1. Gameplay backing: Does not exist. Yuna can always summon an aeon whenever her opponent doesn't have the same aeon out. So we dive into the realm of a pure plot argument.
Nope, the gameplay backing is there, you just skipped over the key part: "
whenever her opponent doesn't have the same aeon out." It's not plot at all. If Summoner A is using Aeon X, Summoner B cannot use Aeon X. This, without any other assumptions, means Aeon X is unique and cannot be used by two Summoners at the same time.
2. Plot backing: Highly tenuous. You are extrapolating from the fact that Bahamut won't fight Bahamut that it is the same Bahamut. This is at leat partially contradicted by the game's definition of an aeon, then, since it's pretty clear that physically they are different beings (constructed from different pyreflies).
Partially contradicted, yes, but not completely because the dual components of an Aeon are integral to the plot (even you mentioned this). If a key component, the fayth itself, cannot be summoned to two places at once, then the summoning fails, regardless if the pyrefly constructions are different. This means an Aeons, in its totality, is still unique and cannot be in two places at the same time.
3. Contradiction with existing DL policy: even if Yuna's Bahamut was Isaaru's Bahamut, it doesn't matter, since no other PC can use Bahamut - the parallel with Ghaleon's Sword and the Dragon Buster and the Bright Shield Rune beat you over the head yet again. We're now down to less than nothing.
All those examples only exist in plot only; they have no relevant impact on gameplay. The second Rose can go around using the Dragon Buster skill to instantly slay dragons, then there might be an argument, but Rose's Dragon Buster weapon acts completely differently than Lloyd's.
Arguments for Aeons not being unique exist in both the gameplay and the plot.