Let's presume a couple things, one which is not quite evidenced in the show (but a fairly logical presumption nonetheless) and one which is foundational to it:
* The prime universe in Fringe is, more or less, our universe. In our universe, it's been widely speculated that there are an infinite number of parallel universes in which history has unfolded in ways which may differ only slightly or very drastically.
* <PERSON> has proven that there is at least one parallel universe, both by observing and visiting that universe. <PERSON>'s experiments have caused our two universes to bond in certain ways which make it easier for us to travel between and interact with the other universe than would otherwise be natural.
Given that there is, for certain, one parallel universe it is then much more likely that the "infinite universes" theory is also true. The universe where <PERSON>, <PERSON>, and friends live just happens to be the first we've discovered. Since it's now also the easiest to travel to and interact with, it also just happens to be the only one we really care to acknowledge exists at this time. It doesn't exclude the "infinite universes" theory - it just means we're currently ignorant to the existence of the others.
Presuming a literally infinite number of alternate universes, the probability of there being a universe in which we see so many significant changes alongside so many significant similarities is just as much as the probability of there being one in which we see the "Butterfly Effect" fully manifested like we'd expect. 283
In fact, by the very nature of the multiverse stretching into infinity, the existense of both these universes alongside ours isn't just probable - it's inevitable.
For example, if our first discovery of an alternate universe were truly by random chance (as, realistically, it probably would be) then it's much more likely that the first universe we ran into would be one in which Earth did not exist at all. Perhaps even the entire solar system, or the Milky Way galaxy, might not even be there. Or we could discover one of a number of universes in which a scientific experiment gone wrong caused the sun to age and cool prematurely, putting the Earth into another Ice Age. Another possibility would be that the first universe we discover is one in which the dinosaurs were never wiped out - instead of humans being the dominant species, the Earth is crawling with these huge, terrifying lizards who may or may not by this time have evolved to have intelligence and technology.
Given all the possibilities, it's really very unlikely that the universe we would first discover would be that one among infinity which is so unlike ours and yet still so similar. But, if it wasn't, then we wouldn't have a show. Or, at least, that show probably wouldn't be Fringe - it'd be more like Sliders.
This may or may not ever be covered in canon, but it's the most logical explanation I've been able to come up with. 283