Atelier Iris 3: Grand Phantasm initially doesn’t appear to have many direct connections to its two predecessors — aside from the presence of the Mana spirits, that is.
There doesn’t seem to be a widely accepted “official word” on exactly why this is, so it’s largely up to interpretation. Some commentators online believe that the game was originally intended to show Iris Blanchimont’s alchemy training, placing the game in between Atelier Iris 2: The Azoth of Destiny and Atelier Iris: Eternal Mana in chronological terms — but either that was never a thing, or it was changed at the last minute. Because, as the game makes clear, the Iris in Atelier Iris 3: Grand Phantasm is not Iris Blanchimont — she’s Iris Fortner.
Ultimately it doesn’t matter all that much; Atelier Iris 3: Grand Phantasm very much has its own story to tell, and has a well-realised narrative setting in which to explore that story. So let’s do just that!