It's established that these take place in the same universe. In "Crossover", Mirror Kira recaps the TOS episode "Mirror, Mirror" and how events played out thereafter:
INTENDANT: Interesting. On my side, Kirk is one the most famous names in our history. Almost a century ago, a Terran starship Captain named James Kirk accidentally exchanged places with his counterpart from your side due to a transporter accident. Our Terrans were barbarians then, but their Empire was strong. While your Kirk was on this side, he met a Vulcan named Spock and somehow had a profound influence on him. Afterwards, Spock rose to Commander in Chief of the Empire by preaching reforms, disarmament, peace. It was quite a remarkable turnabout for his people. Unfortunately for them, when Spock had completed all these reforms, his empire was no longer in any position to defend itself against us.
Yet in "Through The Looking Glass", Sisko seems to think it's just one of many possible parallel universes:
SISKO: If I had to guess, I'd say that this is the same parallel universe that two of my crewmembers visited a year ago.
Furthermore, we saw a huge number of parallel universes in TNG's "Parallels". At the climax, we see:
DATA: Other realties are emerging into our own [...] At this rate, the sector will be completely filled with Enterprises within three days.
And from this question, we know there are at least four different apparently totally unrelated ways to travel between the Prime Universe and the Mirror Universe.
So, if there are so many parallel universes, and the Mirror Universe is apparently one of these nearly infinite number of parallel universes, is it explained what is special about the relationship between the Mirror and Prime Universes that all the different means of travel all connect these two universes in particular?