It's not the same at all, since as far as we know, the Mirror Universe was not the result of time travel. Indeed, it couldn't be. A timeline is not a universe. A universe is a distinct, physical location. A timeline is more like an idea, a concept, a version of events.
But that just means "Mirror Universe" is a misnomer. It has the same laws of physics, the same planets, the same species, the same individuals, the same ships -- only the historical details are different. That can't be anything but an alternate timeline. Something that's literally a different universe wouldn't have any of those commonalities; it would be something totally different, perhaps even down to the physical laws. Species 8472's fluidic space, for instance, is another universe.
Tons of SF shows, movies, and prose tales use "universe" and "timeline" interchangeably. So you mustn't take the terminology too literally. Most of the time, what's called a universe in a story is really a parallel timeline.
And it's an entirely natural reaction to immediately assume that ST XI obliterated existing continuity. In all previous instances of time travel, the new timeline did exactly that: overwrote the old. Now, suddenly, the rules have changed? That's a bit odd, innit?
We don't know that any "overwriting" happened in most cases; it could simply be that the characters moved from one timeline to another. And "Parallels" made it a canonical part of Trek physics that alternate timelines do coexist side by side, so there's definitely a precedent besides the Mirror Universe.