Well, the MU has never been my favorite. It is just a plot device to squeeze several episodes without having to take care of the details in the prime universe, patch things up when a character is dead or put the characters in a new setting.
I don’t think that the characters were exact opposites but rather alternative versions of the same person acting under different circumstances. They simply expressed some latent traits of their personalities that were sublimated or underdeveloped in the PU. Kira’s volatile nature and aggressiveness were transformed into bitchiness, lust and ambition in the MU. Bashir’s eagerness to please everyone and be moderate gave way to his suppressed fortitude and frustration in the MU – he was driven and capable of playing dirty, just like the PU Bashir when he lured and captured agent Sloan and entered his mind. Sisko was still the same badass and charismatic leader. Ezri was a wavering mercenary but still she was analyzing people just like the PU Ezri who was a counselor and she chose to betray the Intendant just like the PU Ezri who joined Starfleet to piss off her bossy mother. Leeta was as lascivious as usual, Jadzia was haughty and frivolous and flirted with the men around her like her PU counterpart. Worf was the same, minus the human upbringing, Garak was using the weaknesses and the mutual animosity between the power structures for his own ends, just like the PU Garak did.
The technology was the same, Vor’cha- and Galor-class ships, the Klingon-Cardassian alliance had the same uniforms. In fact, it never becomes clear how the Klingons have the same ships but without a cloaking device or how the rebels produced the parts to assemble Defiant, having in mind that the Terran Empire did not possess any shipyards or hi-tech industrial facilities, they were enslaved only few months ago.
I guess the books shed more light on this universe but the five episodes in DS9 did not explicate it, they are to be taken lightly, just a side diversion and watching the actors branching out their characters.