I wouldn't entirely rule out a physical mechanism. If you make a big motion (placing it initially or shifting it back into place) then it does not match the floor surface. But as you press down on it in different places over time, it is pressed in. If that makes it very slightly deform, then the contact area increases dramatically, increasing friction to a similar dramatic extent.

Maybe. Depends on the material.

(Note: the approximation that friction is independent of surface area assumes flatness. It does not apply if at least one of the surfaces is deforming to become more conformal)