The way I understand it is that Tiber Septim isn't REALLY dead...it's sort of the same deal as Sheogorath. Each new Emperor MANTLES him and sits on his throne, keeping the power of the realm alive (the "realm" being more the stable Mundus, separate from Oblivion in this case.) The Amulet of Kings is sort of like the symbol of the office, the same as Sheogorath's staff. So it's really like more of an office than a person..the Emperor. Yes, there's a person but that person occupies a place more so than actually being the true authority themselves.
That makes sense, but Sheogorath was a daedra, and it was a constant of him and Jyggalag