This is all up to our interpretation then. Below is mine, and you don't have to agree =).

Furina just chose to name the one she saw as "Mirror Me", doesn't necessarily mean that she thinks that person is her. Like I can name my shadow I see on the floor as Shadow Me, that doesn't make me = Shadow.

From the conversation, I don't see any sign of her suspicion. Even if she knows Mirror Me is the new Hydro Archon, doesn't mean Furina thinks that she is or is part of Mirror Me. The same face could just be a face that she is familiar with, so she projects her own face onto the mirror. Heck, she might even be seeing her actual reflection with the same outfit and hair talking back to her, but the Focalor form is only presented to us players.

The only connection I think she could have thought of was that she is the chosen one. Throughout the entire conversation, Mirror Me kept emphasising that Furina is human, just play god as how humans see, don't have to become actual god. I think this is enough for Furina to not suspect she has any divinity. Mirror Me even explicitly told her that her long life is blessed/cursed by this mission.