The Trio of Elemia are actual mythological figures in Ar Ciel that existed far before the Towers were even in planning stages: they are the protagonists of the first myth to ever have been told in that planet (Stelliarhythm - Festival of the Stars), and they were the first three demigoddesses to have been created by the Wills of the Planet shortly after humanity was created. Each of the three sisters had power over an element: Eolia could control the wind, Frelia the earth and Tilia the water; and since they were what gave hope to the newborn humanity and allowed them to continue exist, they were revered ever since. This is also the reason why the three Origin Reyvateils are named after each of the three mythical sisters: Shurelia (true name: Eolia), Frelia and Tilia.
However, the Tilia that appeared in the game was no more than a disguise Mir used to get Aurica and Lyner to cooperate in the plans for her resurrection, and Aurica has zero relationship to Mir aside of them both having similar enough mental wave frequencies that the Tower mistakenly assigned her a copy of Mir's Hymn Code. The only reason Mir says that is because she considers all Reyvateils except for Shurelia and the Star Singers as her daughters during her hatred-filled phase. Mir already grew out of it, as a comment she made in the official fan-site Ar Portal reveals she doesn't want to be associated with Aurica in any way, and even feared what would have happened if she ended having Aurica's personality after she took over her.