It is unlikely in the extreme that the MU Enterprise would contain mostly the same crew and all the command crew of its prime universe counterpart. Having said that, however, it was the only universe with which the initial crossover could have occurred. Or I'm just plain wrong and many cross-overs took place and the events of "Mirror, Mirror" took place across a multitude of alternate timelines. I have heard, but am unable to substantiate, that the original plan was to imply that Kirk and company had not
returned to their original universe at the end but that idea was dropped in favor of the Marlena ending. (As an aside, I'd love to see a follow-up involving prime universe Marlena and Kirk's "observation" that they could become "friends", knowing what Kirk's definition of such a friendship would doubtless entail.)
The idea that there is a multitude of MU-type alternates is sort of already out there, what with the comic stories, Dark Mirror, and the other various lit efforts.