Unfortunately, this one actually sounds plausible.

Colt is obviously human, we've all seen it in TOS. So if Colt is now an alien, then how can DSC be prime?
Because fiction is just a story being told, and it doesn't follow the same rules as reality. You can change something in a story after the fact and still pretend it's the same story; the new version just overwrites the old version. The current Marvel Universe in the comics where Tony Stark got his injuries in Afghanistan is treated as the same reality as the '60s Marvel Universe where Stark got his injuries in Vietnam. The later TNG where Data doesn't feel emotions or use contractions is treated as the same reality as season-1 TNG where he did both. The
Planet of the Apes movies where Taylor was said to have fallen through a time warp pretended to be the same reality as the first film, in which the astronauts explicitly were in cryogenic suspension for 2000 years. The details are changed but the pretense of continuity remains, because this is all just pretend anyway, and rewriting and changing one's mind are fundamental parts of the creative process.
The only thing that's required for two stories to be in the same reality is that they
pretend they are, even when they disagree on details. Because "realities" in fiction are nothing more than story devices to begin with. If you can buy the fiction that the story took place at all, if you can suspend disbelief about the obvious false claim that we can see events centuries in the future, then you can buy the fiction that a previous story happened differently than we were told the first time. It's only an "alternate universe" if the story
says it is, because the story's needs are what shape the reality within it, not the other way around.
Besides, as I and others have been saying, there are numerous creative ways to reconcile this without needing alternate realities. This could be a different character also named Colt. It could be that Colt was transformed in some way in the 4 years since "The Cage" (if Airiam could be a transformed human, then why not?). For that matter, it's never explicitly stated in "The Cage" that Colt is human, just implied because she's chosen as potential breeding stock for the Talosians' attempt to create a human population. But then, maybe they determined that her species was genetically compatible with humans and that her attraction to Pike was more important. As for why she looked different, well, Robin Curtis looked different from Kirstie Alley, and Anson Mount looks different from Jeffrey Hunter. Visual depictions can be disregarded.