In the first Deadpool movie, Deadpool gains his powers as part of attempting to force a mutation. And he does get a mutation: the incredible healing factor. He also gains the ability to break the fourth wall and recognize he's in a comic book movie some time shortly after the mutation.
However, neither movie is really clear on whether his ability to see and interact with the audience was a literal mutant power, a side effect of him going insane from the therapy, or just a running joke that isn't bound by logic.
After reviewing the script again from the advice of a friend offline, it looks like Wade's 4th wall breaking isn't a power. While he has his collar on, he quips the following while turning and looking directly at the camera:
Fun fact about the Ice Box... though no one's ever seen it, they keep a monster in the basement. Right next to a huge, steaming bowl of foreshadowing. https://www.springfieldspringfield.co.uk/movie_script.php?movie=deadpool-2
So, even when he's powerless, he can still call out movie tropes and recognize that it's a popcorn flick with an audience.
It's not a mutation. It's actually a sign of his madness. Deadpool gained his powers through torture1, which mentally broke him.
Not only is he under the "illusion" (in-universe this is seen as a derangement) that he is a fictional character (his frequent 4th-wall breaks)
Ryan Reynolds has told Empire that he’s insistent on Deadpool‘s habit of breaking of the fourth wall to carry over from the comics to the big screen. That means, in effect, that Deadpool/Wade Wilson will sometimes address the audience directly, as with Ferris Bueller, say, or the character of Paul in Funny Games.
He also suffers from schizophrenia (voices in his head)2,
And while it could be attributed to the excessive amount of pain he undergoes on a regular basis - his reaction to pain and being mutilated over and over again is not one you would expect from a sane person.
1. In Deadpool 1, Ajax explains that the Weapon X program requires the body to undergo increasing amounts of stress in order to trigger the mutation.
2. This is in the comics, but not depicted in the movies. It is possible that this was tied into the 4th wall breaks as talking to the audience as the "voices" in his head
Movies are a little more vague on this but if you include comic lore it is much clearer. So I have read a lot of Deadpool comics and I can tell you he is fully aware that he is in a comic book but it is not derived from his mutant powers whatsoever just a side affect of his craziness. At one point he instructs the readers to update his wiki page to show that he was offered to join the X-Men and refused after Storm asks for his help even.
If you are looking for a character whose power revolves around manipulating the comic book she is in I would recommend the Gwenpool character. The Unbelievable Gwenpool #13 even has Deadpool and Gwen talking about being in each others books.