This is where the question of what sci-fi is comes into play.
Stepping away from that for a moment, the issue is that if we said, "Kirk has still become a captain in this universe, but on a totally different ship with a totally different crew," or "Kirk is a restaurant owner in this universe," we would have a very different story, though not necessarily better or worse, just very different. The idea of a very similar, other universe has to be accepted to tell the story the writer apparently wanted to tell.
So is it sci-fi or fantasy and does it matter? In my opinion: Firstly, if a person likes the story, it may not matter to that person; secondly, in my view, we can look at it like Joe Jennings said in an interview about ST:II. He said that in sci-fi you need to create your own rules and be honest to those rules. Does "Mirror, Mirror," do that? If so, it is sci-fi to me. In my mind it does, since the "rules" are that a storm and a transporter accident got them here, and that if they survive they have a short time to get back to their home, and in the meanwhile they see similar, but evil, versions of their friends.