Yeah, it seems the Master Sword has some sort of conscious tied to it. Even in OoT and TP that was the case. But I really doubt it is Fi, unless she lied about her conscious dissolving away.
Fi's conscious began to eradicate along with Demise's. She can't be part of the Master Sword any longer. And by her saying maybe they will meet in another life, that implies she too would be reincarnated, but not into the Master Sword once more. Maybe Navi is a reincarnation of Fi?
This is an...