I've seen fan sources assuming that Spock's 11 years, 4 months, and 5 days under Pike's command encompassed two 5YMs and a bit more, but nothing in the tie-ins seems to support that.
Although let's define our terms a bit more carefully. The phrase "five-year mission" by itself is pretty meaningless. Duration doesn't define a mission; purpose does. What's new as of STID is a dedicated deep-space exploration mission with a planned duration of five years. Which strikes me as a reflection of the particular state of affairs in the alternate reality, where Starfleet seems to have adopted more of a defense-oriented footing in the wake of the Kelvin's destruction. Perhaps it's only now that the pendulum has shifted back toward Starfleet's exploratory responsibilities. If so, that doesn't reveal anything about the state of affairs in the Prime reality.
Well, whether that's really new as of STID depends largely upon whether one considers William Shatner's voiceover in TOS to function as words from the mouth of an omniscient narrator, speaking of the way things are in-universe. If you do (and I do), then it's reasonably explicit that the TOS Enterprise
had been sent by Starfleet on a mission that had a planned duration of five years and that had the intended purpose to explore strange new worlds, seek out new life forms, etc. Perhaps you are arguing that the TOS Enterprise
's flight path wasn't always dedicated to traveling in deep space, since she often returned to base, or that the mission was not one exclusively of exploration (both of which I'd concede)?