There's nothing in the movie that says it has to be "our" Old Spock.
The fact that he says "I have been, and always shall be, your friend" is pretty telling. Would a Spock from an alternate reality have chosen his words the exact same way?
Besides, I'm speaking metatextually. The reason the filmmakers cast Leonard Nimoy -- coaxed him out of retirement, even -- and built the story around Spock's return from the 24th century was to make this story a legitimate continuation of the Star Trek
that had come before rather than a completely disconnected version. If that hadn't been their intention, Nimoy's Ambassador Spock would not have been in the movie at all.
In the Mirror Universe McCoy's work table had the exact same acid stain as the one in the prime universe. That seems much more unlikely than someone using a particular turn of phrase. and yet there it is.
An actor "un-retiring" is hardly unprecedented and Nimoy has done it before and since. Give him a big enough part and a big enough payment and you could convince any number of retired actors to put on the tights one more time.
The writers intent may have been that it was "our" Spock that we were seeing. In fact, I'm confident that you could say that with certainty. However, just like TATV being a holodeck program gave you an out for undoing Trip's death, the fact that the movie establishes itself as an alternate universe allows for the interpretation that this is a Spock from a universe very similar to but probably different in some ways, to the one we saw on TOS, TNG, etc. Nothing in the movie itself prevents that interpretation, writers intent being worth nothing more than their opinion. If it's not on the screen, it doesn't count.