Physics-defying fortress in the sky? Or mirrors-for-tiles?

I prefer the former.

If it were mirrors, then we'd see the underside of the girl and the chair. Which would be awesome to invite girls in skirts over.

A flying fortress with glass tiles for a floor would be nice

Nah, too close to being a "glass house" for my comfort.

Where does the assumption of mirrority come from? They clearly aren't reflective. So it's just blue tiles with cloudy pattern on them, simple as that.

It is reminiscent to a few scenes of the enemy airship in Nadia, where they did either have a glass floor or screens in the floor that showed what was below them.