As I said, the set of reachable universes is not infinite. Remember, we're talking about two separate explanations for parallel realities. One is Many Worlds quantum theory: that our own, single universe is constantly branching into multiple parallel histories, all linked by a common origin but having events unfold differently following the point of divergence. (Which can be a divergence caused by time travel, like the Abramsverse, or a spontaneous quantum branching, like the timelines in "Parallels.") In that explanation, it makes sense that different timelines would be connected and potentially reachable, because they literally occupy the same physical space, just out of phase with each other. And their similarities are explained by their shared origin -- which means they can't have fundamental divergences like different physical laws.
The other is the random-chance argument: that in an infinite multiverse containing an infinite number of universes, every possible universe would inevitably exist somewhere, so duplicates of Earth and humanity and ourselves as individuals would just randomly happen to exist in those universes, despite the immense improbability of such duplication by chance. And this model is also often used as a rationalization for alternate realities that can't be explained as parallel timelines because they have different physical laws or different planets and alien species, rather than simply different histories. And my objection to that as an explanation for alternate universes in fiction is that those universes, while they might technically exist in principle, would be unreachable due to the infinite time it would take to search through all the infinite universes and find such a duplicate. (The argument is not that any other universe would be unreachable, but that those that coincidentally happened to duplicate ours would be infinitely outnumbered by those that were profoundly alien in every way. Although any of these other universes would probably be at an inconceivably large physical distance anyway, and thus unreachable in that sense too.)
So the idea is that, since the Mirror Universe is reachable, it must logically be a divergent timeline of our universe, with its similarities arising not from random chance but from a common origin with our universe. The interpretation from the Abramsverse comics, that it's just one randomly occurring duplicate out of an infinite set, doesn't really work from a mathematical standpoint.
Now, that's where I can see the appeal of the "everything happens in infinity" model, because it allows justifying even the most bizarrely unlikely of coincidences. But it's hard to reconcile with the easy reachability of the MU. Perhaps the answer is some blend of the two: Out of all the different parallel timelines branching off from our universe, which is a finite but still very large set, the MU just coincidentally has a lot of the same people despite having a very different history.
Although the problem with that take is that most of the other alternate realities we've seen, in "Parallels," Myriad Universes, and the like, also tend to have the same people in them, often in similar roles. So maybe there's some quantum-probabilistic resonance between timelines that causes similar events to occur and the same people to be born. There's a case to be made that most timelines tend toward the most probable course of events and thus would be likely to converge in certain ways (although one would expect that it would be larger-scale events that would converge while smaller-scale events like who gets born or who gets what job would be different).
Or -- here's a thought -- maybe there's some quirk of quantum probability that means a given individual whose worldline intersects with other timelines is most likely to intersect those that have other iterations of the same individual. So it's a selection bias: As long as we follow the adventures of certain characters, we'll only see them visit those alternate realities that have their counterparts in them, or have had them in the past.
