As anyone who watched Prodigy knows, the main characters + Chakotay end up on the ISS Voyager A during some accidental multiverse hopping due to time shenanigans.
It could be argued that the mirror universe is just a random variant of it in the multiverse where the Terran Empire is still prominent.
However, I noticed two things: Terran Admiral Janeway refers to the fleet as “The New Terran Fleet”, maybe suggesting a success for the Terran rebellion.
This is further supported by the fact that her combadge is the same as (and her uniform plausibly an evolution of) the ones seen in the various IDW mirror universe comics, the earliest of which is from 2017.1
The plot of these comics tries to fit in with the info from DS9, revealing (in its own continuity, at least) that the Terran Empire still existed during DS9 but had been reduced to the Sol System, with most people outside it not even realizing it still existed. The comics chronicle the resurgence of the empire and the rise and fall of Picard.
Is it possible that the timeline we see in Prodigy is actually the same mirror universe in DS9 and that the IDW mirror universe comics have been semi-canonized?
1: Coincidentally but not relevant to the question, the ISS Cerritos shown in the holodeck in LD:I, Excretus also uses these badges, but not the typical mirror universe uniforms, although it could be those are just the mirror Cali Class uniforms.
On that last bit, I guess it has precedent. For instance, the look of the Titan on LD is largely the same as the design shown on the cover of some novels, but Riker’s bridge officers are different.