Although I'm no fan of the "infinite universes" model, one could legitimately argue that even in a large but finite array of universes, there'd be a nonzero chance of finding two in which the same individuals existed. There could be a few thousand alternate versions of the Mirror Universe, and in most of the ones where Betazed and Andor were razed, the ancestors of Deanna and Shar might have been killed and they wouldn't exist; but we preferentially follow one in which they do exist, and in which most of the other Prime Universe characters do as well. Maybe that coincidental convergence of individuals is why a link was established between the two timelines allowing recurring interaction between them.