The Ivalice raids in FFXIV are no more a sequel to FFT than Crystal Tower being a sequel to FF3. They're just homages. If anything, it's heavily implied that Ultima was a primal/eikon long ago, and Ultima Weapon was an attempt to reverse-engineer Omega's technology, but they named their new creation after Ultima, a legendary savior/angel. It's also implied that Ultima's power/domain is not holy/light but rather illusions. Her ultimate attack in the raid in the English localization is "Ultimate Illiusion", but in JP it's simply "Final Fantasy". It's likely that the black magic spell "Ultima", which Ultima weapon used to annihilate the Praetorium was based on recreating Ultima's spell. The original spell that Ultima casts might be an attack that draws on what the target might imagine the most destructive spell to be like. Similar to Aizen from Bleach who can manipulate the 5 sense of his target, Ultima's spell casts an illusion to make the target imagine like they are getting hit with the most destructive spell of whatever their mind can conjure it to be. Ultima's powers seem to revolve around trickery and deceit. She creates Auracites which are fragments of pure power. This attracts mortals not unlike how the sweet smell of nectar might attract flies to a venus flytrap. Mortals often harbor deep desires and vices, which is what Ultima feeds on. Mortals get the auracites because they believe they grant power to fulfill wishes. In reality, the stones resonate with whatever emotions the user has, and amplifies them. This is what Ultima wants. When the Bangaa takes the Duma and becomes Yzmat, he harbors emotions of vengeance and becomes a massive dragon. Ultima feeds off this like primals feed off prayer. However, the caveate is the stones can also grant non-evil miracles to wielders of the stones whose intentions are pure. It's sort of an unintended but low-probability scenario from Ultima's perspective though, since she knows the ones tempted to wield the power of the stones are inherently greedy, corrupt, and fowl, and that's what she's counting on for sustenance.
But yeah, the spell named Ultima is a long-lost myth generated after the Primal named "Ultima" and Ultima Weapon was the name chosen for the reverse engineered Omega technology, named after the goddess of lore from Ivalice (who were the ancestors of the Allag).