Couple possibilities:
- The Punk is the "same person" in so much as he is a parallel universe (MCU) version of the Star Trek universe version. Punk on Street may or may not have been Vulcan nerve pinched in his youth. The MCU is quickly delving into their own Multiverse, and the Star Trek universes may be their own thing, where Nick Fury fought in the Eugenics Wars and superheroes never rose up due to the lack of an Iraq War (in the 2000s) or something. How to consolidate Marvel Cosmology with Star Trek's is the biggest concern, as I don't think we can assume all of the MCU adventures take place out of the Galaxy (the simplest solution).
- Kirk and co. did not actually travel to their own 1980s, they traveled across space and time to a (duplicated?) Earth in a much different galaxy (to account for the cosmos differences). Star Trek IV is the first movie (partially) set in the MCU, even though it begins and ends in the STU. Scotty giving away the formula for transparent aluminum is no big deal, since he's out of his universe and SHIELD probably already has that stuff at the Triskelion thanks to Howard Stark and Hank Pym.
The universes are grossly incompatible the more I think of them. Perhaps, maybe, the Whale Probe was somehow involved in sending the Enterprise (during their LSD trip) back in time through the Land of Fiction. Or maybe Kirk and co. we're all just hallucinating based on the movies (which the Probe, picking up Earth signals, was deeply familiar with), and the whales were just replicated for them.
I wasn't being totally serious in my last post. The cameo did indeed happen, but as you pointed out there's more than one explanation for it. Though, there are some things that actually make them being the same universe more possible than they seem at face value. For instance, the Skrulls, Kree, and some other aliens are actually from other galaxies and some combination of SHIELD, Hydra, and other groups could be the origin of Section 31. However, again, as you pointed out, there's still too many differences for them to be the same universe. The most likely possibility is that they're parallel universes which share some elements of history, but also diverge significantly on other important points.