This has been discussed numerous times, and AFAIK it was deemed impossible due to the Gamebryo engine
Its not a specifically "gamebryo" limitation. The engine has to be coded for it, and given the graphical demand of modern games, mirrors are not a high priority to mess with.
Duke Nukem 3D had mirrors, and that was 15 years ago.
That was back when the idea was new and everyone wanted to experiment with them. They have since disappeared as the performance impact is too massive to merit full implementation.
Funny thing is, that is how Oblivion's water reflections are done. Objects in the reflection are drawn twice. So really it is drawing everything three times. A mirror would be a far less performance hit than say Lake Rumare, especially if it only drew everything in it twice. Making a clone of the PC is also simple enough to do. Oblivion is already rendering you every moment of the game. Making a duplicate of something is simple in the digital world. In fact, Oblivion does it when it makes your statue at Bruma, and Shivering Isles does it as well as one of the opponents you must fight (though it is a dark clone). Mimicing your movements falls under making a duplicate. Again, simple. It is nothing but a series of animation files that the game plays. Instead of kicking them off once for one actor, it kicks them off twice for two actors.
Thats a bit different. Water reflections are kindof like mirrors. They do indeed rerender the landscape. They can render trees, and even the player, but the more you increase what is reflected, the more demand. So water are like limited mirrors. They render only the basics to get the point across.
Shivering isles just duplicates your character graphic. To make a mirror you would need to sync the animations and ensure there was no delay between when you moved and when your reflection moved. This is where the performance impact kicks in, as having to update a clone to mimic your movements every frame is very demanding of a process. Now toss on 2 or 3 NPCs, and the demand just goes up exponentially. And got help you if there are items laying around that have havok. Static models are much much easier to duplicate then NPCs and objects with physics.