It's basically a dyson sphere ship.
It's not exactly a stretch to think it's possible. The Federation isn't a military dictatorship. The entire economy of the Federation isn't subject to the whims of the military. The Terran whatever however is a military dictatorship, and every single dime can be squeezed into military based R&D.
The whole discovery spore drive thing, can be pretty easily explained as experimental technology that is going to end up being too dangerous, both to the crew, and the universe at large, and the whole thing will eventually be black boxed and classified. Which is why there is no mentioned of trans dimensional FTL travel in any of the more modern shows chronologically.
I'd honestly be kind of surprised if they keep the whole spore drive thing going beyond a season or two (if the show makes it). The only way I can see them keeping it going, is if they are stuck in the mirrorverse for the entire duration of the series, and the show ends when they get back. It's clearly detrimental to the crew, and the whole plot now seems to be that using the mycellial network is damaging to the fabric of reality.