Buuut... At the end of OoT, Zelda taks the ocarina back and sends him back in time, basically to his old life. Later in MM when he gets the Ocarina from Skull Kid a memory goes through his mind.... A memory that, according to the gameplay of OoT, never happened! So How did he get his Ocarina again?
That event took place sometime between the final scene in OoT and the first scene of MM.
OT: The eye argument is shattered once you realize that the fused shadow only has one eye. They are not the same. I think the one eye creates a stronger connection to the Shiekah than to Majora.
The Twilight Realm is not a parallel universe. It is a separate realm created by the light spirits. It certainly has nothing to do with Termina, as Termina had been around for a long time before MM and the TR was only created shortly after OoT.
I see Termina not as a parallel universe created by alternate decisions, but as a separate reality that exists 'alongside' the familiar one which has certain similarities to it such that they could be considered 'parallel.' (If you think of time in three dimensions, the first being the progression of time in one universe, the second being parallel universes caused by alternate decisions, the third being alternate realities which exist independent of one another (separated by 'void*'), Termina's relation to Hyrule could be represented by a displacement along the third (z-) axis.)**
*borrowed from Doctor Who. I think everything we see in MM from the hole with the glowing symbols up to the twisted path beneath the clock tower is part of this void.
**"This is, of course, impossible." - The Hitchhiker's Guide to the Galaxy, on "Milliways, the Restaraunt at the End of the Universe" [aka fanfic, not based on real science, I'm not qualified to think about such things, etc.]