The existence of Fantasia is not acknowledged by dialogue, so it has no continuity with any other lore.
Outside of Fantasia, there are some cases of permanent changes to a being's physiology, but none of them appears to be fully controllable.
One tempered by a primal would take on features of that primal's element, becoming monstrous. Ishgardians who partake a dragon's blood may find themselves transformed into a dragon, usually one of three types. The Mad King Theodoric transformed his rivals into monsters. And the pixies also like to transform those who wander into Il Mheg—though other than leafmen, the results aren't fully predictable.
There are also targeted transformations used as temporary enfeeblements: adventurers may find themselves transformed into a toad, an imp, or an old crone, though usually they turn back before long—assuming they survive.
Finally when it comes to glamour, we as players use it to make one item appear to be another of a similar size and shape. But other users of glamour magick are not always beholden to such limitations.
The false Inquisitor was able to glamour himself into a dragon much larger than he is. And the sylphs are adept users of glamour, able to pass themselves as one of the five races, and even mimicking the Warrior of Light in the tribal quests. NPCs in the ninja quests also wield ninjutsu that allow them to pass as someone else; one antagonist scales himself up to face you.
Such uses of glamour do not seem to be common.
Nanamo doesn't use glamour to become her alter-ego, I wonder why?