The parent commenter means to say that for most of the game in Deus Ex, the player doesn't see the protagonist. (Can't remember if the cutscenes shown him from a third-person view.)
https://m.youtube.com/watch?v=jnbCwPwJNFY&t=3m30s
So maybe it was the first player-controlled viewing of yourself.
I kinda miss stuff like that.
Sadly, when things break like that now (looking at you, Bethesda) it usually leaves the game in an unplayable state, versus reverting to a known-good state.
Modern games also do a lot of tricks when it comes to mirrors. Real reflections are expensive. Though they're becoming increasingly common thanks to the ray tracing hardware on recent graphics cards.
As a player I recently noticed that in The Last of Us Part 2, when you're in front of a mirror the windows give off an unnaturally bright light to hide that the outdoors area wasn't reflected.