You can say their separate realities to justify the mess but its not the case, the last Star Trek movie is supposed to be about to how Kirk became Captain of the Enterprise not some alternate reality and it can't be an alternate reality anyway cause you have the Spok we know as well as the new one so it has directly got to do with temporal time travel.
If you have ever seen the original pilot episode of the original Star Trek with Captain Pike you would see how hypercritical the whole Star Trek story line has become and there is another Episode with Kirk returning Pike to the planet it all started with but the new movie completely wrecked that whole story.
The words "alternate reality" are used IN THE FILM.
Spock chased Nero back in time. They landed in an alternate reality that was changed because of Nero destroying the Kelvin. The theory is that going back in time CREATES alternate universes.
From there, it's the story of how an alternate Kirk comes to command an alternate Enterprise.
And our Spock is there because it's HIS black hole that created the alternate reality by sending Nero to change history. And the running conceit of the movie is that changing history creates alternate universes. (Which is one scientific guess about how real time travel would work. It's not USUALLY how time travel works in Star Trek but a lot of sci-fi suggests that your method of time travel determines how the rules work, including whether you can alter your own past or whether any attempt to do so creates an alternate reality.)