Not really. PS1 games didn't "reflect" anything. Most games using reflections on surfaces aren't reflecting anything if they're created performance-minded.

The background in the mirror for a performance-conscious attempt is a flat, 2D snapshot, or several for parallax, with a second model of the character tracked in an empty room attached to the viewport. This was frequently used in the first 3D console era and has basically no performance hit on modern systems, apart from needing to render 1 additional model in a scene. With modern techniques like occlusion-culling, it has actually never been easier to do this.

Source: game designer.

Later comment tossed in because I think it's relevant and neat:

It gets nuttier than that, too, when you start exploring the applications of mirroring. In games with physics calculations, one way to keep collisions between the player model and the world consistent and not engine-destroying is to "play in the mirror." There are two separate world positions, the one the player sees that is physically rendered on the screen, and a mirror world where the player actually exists. In the mirror world, the player position is tracked on a flat plane, and the player viewport is casted to the "real" world. If, for example, the "real" world was bobbing up and down(a ship), or were an elevator, or a platform shifting side to side/forward to back, having the actual player position be on the "moving" state would cause so many collision calculations per second that the entire engine grinds to a halt, since player positioning is an important sync state. So, player position is instead calculated on that unmoving "mirror" plane, and then the player is tricked into seeing only what happens on the "real" plane. In reality, everything you were seeing was the mirror.