Excluding the narrative function of the mirror universe, does the alternative timeline established by the Abrams-movies relate to the "regular" TOS universe in the very same way the mirror universe does? Are those two - on a structural in-universe level - insofar identical as both of them are structurally equal branches which portray "what if"-situations that may have become the "standard reality" if something else had happened at a certain point in time?
Not really. The new Abrams Timeline is identical to the old one up until Nero returns, which marks a definite and catastrophic split between the two. After this split, events proceed extremely differently.
On the other hand, the "Mirror universe" and the regular one do not seem to have a definite "splitting point:" the two are wildly different, but they have existed parallel to each other for millenia, as evidenced by Phlox's references to classic literature being "more light-hearted" in the regular universe in ENT: In a Mirror, Darkly. Nonetheless, thousands of years later, we have a Benjamin Sisko living near the Bajoran wormhole. Seems like an awfully large coincidence... unless something else is going on.
Forgive if this sounds absurd, but I've always held the view that the mirror universe was a special example of an alternate reality that was the "mirror" of the regular one, not because of any specific event that divided the two, but because of some peculiarly of the space time continuum that linked the two together.
The new timeline is certainly a parallel universe, but there are lots of those. It would be inaccurate to describe the new timeline as a "mirror universe" however, as that term hold a special meaning in Star Trek.
I keep asking myself, Why didn’t Abrams hire Mike Okuda to check his continuity? Or ANYBODY, for that matter, that was actually familiar with the canon. It was a big budget movie and they couldn’t hire a consultant to tighten up the plot for the trekies. For example why wasn't the Kelvin in the first movie a Daedalus class ship with the the crew wearing the old uniforms from the Menagerie pilot episode? Instead Abrams just made up a new ship class! After the "Nero encounter" the timelines changed, so Abrams could then "make up stuff" without any issues. But, he chose instead to do nothing, why not put in a small bit of continuity? Is it really too hard to throw a "bone" to the people that have paid money so far to keep the franchise alive up to the point of the first Abrams movie?
I think in reality, behind the scenes Abrams and the other higher up folks at Paramount had a good laugh at the trekies, by not bothering to deal with the continuity in any way. I just don’t undertsand how all this happened, especially in an age when directors release “special directors cut editions" with all the nerdy stuff (like Watchmen for example). The old Trekies didn’t even get that "bone" thrown to them.
However with all this, I was thinking that we should just consider the Abrams Star Trek the “Mirror Universe” told as revisionist history. Once the Abrams version is no longer “Hot Stuff” hopefully a writer knowledgeable about “Star Trek canon” can come along and upend the whole story and say, “well, that was the Mirror Universe and nobody told you so at the time”.
My point is, we should just accept this universe to be the "mirror universe of the Terran Empire". It makes all the continuity issues more tolerable:
1) This would explain why the USS Relativity doesn’t show up to fix the timeline. Because it’s the mirror universe and the future Federation doesn’t care about what happens in the Mirror Universe timeline.
2) This would explain why there is no George Samuel Kirk. The Mirror Kirk may not have had a brother in the Mirror Universe, so he is essentially George, but is named Jim due to the destruction of the USS Kelvin and the early death of his father Lt Kirk. It also explains why he is such a trouble maker without conscience, unlike the original Captain Kirk. Mirror Kirks background was never discussed, so to a certain degree anything goes.
4) This would also explain why Uhura and Spock have a relationship. This Mirror Spock doesn’t care about his bond with T’pring and with the destruction of Vulcan she dies anyway. Not the best explanation in the world, but its better than just outright ignoring the Amok Time episode.
3) Why does Pike not get court martialed for doing such a poor job defending the planet Vulcan and directing the Federation fleet? Because he is an officer of the Terran Empire, not the Federation. To some degree the Terran Empire doesn’t care otherwise, whether Vulcan survives or not. Since the Romulan Empire is independent in the Mirror Universe, the issue of defeating a new powerful Romulan Ship is a more pressing matter than saving Vulcan.
4) It would also explain why the USS Kelvin looks so weird. The design of the Kelvin that we see in the Abrams film could have been influenced by the appearance of the more advanced Constitution class USS Defiant, which appeared in Tholian space during the time of Captain Archer from the Terran Empire, not the Federation. This could explain the earlier diversion of the timeline that precedes Nero and Narada and makes everything so radically different (i.e. technology differences, earlier ecounters with the Cardassians and the existance of nokia).
5) It was well known that Khan was despot in the 20th century, why would the anyone in the Federation be interested in working with him? They wouldn't and that shows its the Terran Empire at work here. Also why do Starfleet Captains need to meet to catch a criminal/fugitive? Isn't that job for Federation law enforcement? More clues that says "Terran Empire, Mirror Universe"
6) Actor Simon Pegg (Scotty) recently said that they may be the Mirror Universe Crew. Why would he say that? I think because like us, he actually watched some of the Star Trek series/episodes in his lifetime and has had a reasonable response as to why nothing in the scripts match what has already been written. He can't just come out and say" I watched enough of the old Star Trek episodes to say with confidence that my boss is an doesn't know anything about the old shows and is writting a lot of "bunk" into the new Star Trek Scripts"
I think it will be a "cake walk" for writers to retcon this version of Star Trek in the future. If the Tholians could open a gateway into an alternate future universe, then the Red stuff that Spock was using could do the same effect into the past. In fact this timeline could be an attempt by the Terran Slave resistance to keep the Terran Empire from losing power to the Alliance with the assistance by maniputating the time line in the Mirror Universe behind the scenes.