Why does Flare have the incorrect cast aura for its damage type?

[ReynTime]

It's mechanically important for BLM.

Flare being Fire elemental means that it does 1.8x the listed potency while under the effect of Astral Fire III and 0.7x the listed potency while under the effect of Umbral Ice III. This means that even though the listed potency is 260, the two Flares in a post-68 AoE rotation have potencies of 182 (UI3) and 468 (AF3), an average of 325.

If Flare were non-elemental, it would always have potency as listed and may be overpowered for an ice phase or underpowered for a fire phase--worst cases, it would never get used because it's too weak to compete with Freeze spam or it would be spammed as either Flare spam or Freeze-Flare alternating. Being Fire elemental allows them to set a potency that self-adjusts to be appropriately low or high to match the BLM's current element. Foul does not need this self-adjusting balance because it's on a 30s charge timer.

That makes more sense.

[Vyrerus]

Flare is fire elemental in XIV? I mean, technically speaking no action has any elements in this game, seeing as they got rid of elemental strength/weakness over 7 years ago.

Every spell in the game has an elemental aspect, with the neutral aspect being referred to as, "Unaspected." There's no profiting off of these aspects, even in areas where elemental weakness are a thing, like Eureka, but they do exist, nonetheless. One place they do feature in, albeit in a minorly important way, is in the Masked Carnivale. The BLU spell list is full of spells of different elemental aspects, and monsters in the Masked Carnivale often have ye olde elemental weakness that you can profit from.

As far as BLM goes, it's what Rongway said already. BLM deals with 4 aspect types in their spells. Fire aspect, Ice aspect, Lightning aspect, and unaspected. Thunder spells and the Unaspected spells may be cast while in the ice or fire phase, and will still deal their full damage as well as cost the same MP/resource always. They could give BLM many other elementally aspected spells if they wanted to, but they stick to those 4 because they are the 4 original magic types relating to damage from BLM's first appearance in the series at large.