I don't think there was ever consistent lore developed by Westwood for Kane's origins and nature. His final pose in the ending FMV seems more like a nod (no pun intended) to the original C&C's GDI ion cannon ending, while the missile itself is a kind of downer ending for the player ("You just blew up the Earth. Have a nice day!"), and at the same time an unexpected surprise because the Nod campaign narrative in TS went away from portraying the Brotherhood straight out evil to something more of "they have their own cause to fight for", only to find out that they were ultimately manipulated by Kane to wipe out humanity (apparently).
The Nod ending in TS was probably not intended to be canonical for the planned sequel so they just went with what seemed cool I guess.
Also in C&C Renegade (IIRC), Kane almost straight out states he's the biblical Cain, when the player discovers the tomb of Abel under the temple in the final mission.
BTW, it's your best explanation of why Kane is immortal, and you could give it a sci-fi twist in the style of Stargate if you wish.