This is an interpretation based on actual facts from the movies, you can recheck them and burn me in the stake if I'm wrong, but please do it so based on similar data.
Project PEGASUS might have the same name and be in the same facility, but in Captain Marvel is NOT a SHIELD Project.
1. In Captain Marvel Movie we see that:
SHIELD is aware of Project PEGASUS. In the movie Nick Fury says Pegasus location is classified. However, his low security clearance is enough for him to know where it is.
At the gate he identifies as a SHIELD Agent, he is not saying "this is Agent Fury" he is saying: "Nicholas Joseph Fury, Agent of SHIELD", this may be an overstretch but one can imply from this that he is from an agency that is external to PEGASUS.
It's not easy to get a picture, but around minute 43 they arrive at the actual base and we see in the screen this title:
JOINT NASA USAF FACILITY
Not SHIELD-NASA, but USAF-NASA
When they're in, there's not a single SHIELD scientist or agent in sight and they're received by military personnel. Then they are being kept waiting and decide to look for themselves, that's when Fury sends a beeper message to SHIELD asking to come and help them. If this were a SHIELD base they would be surrounded by agents in less than 5 minutes, since... it's a SHIELD base. But they're not.
One can also think that being a SHIELD Project there's no need to go look for intel since they already have it... unless this is a top secret undercover HYDRA base that nobody knows about (like the one in Captain America: The Winter Soldier) or from another entity (like the Guesthouse in Agents of SHIELD). But this is a simple government research facility accesed also by NASA which is already known to a Level 3 Security Agent like Fury.
2. Differences with Avengers Movie:
Another detail is that in Avengers, Project Pegasus is oficially looking for new energy sources (though they're really making weapons, as Captain America discovers later in the movie), while in Captain Marvel Project Pegasus is developing new aircraft (which makes more plausible the participation of the US Air Force).
3. The Avengers Prelude
My take, based on The Avengers Prelude: Fury's Big Week is that this project was ditched by the Air Force, taken by the World Security Council and reassigned to SHIELD, with an entirely new objective (they even threw away the logos).
Other Sources that may support my observations:
mcufandom wikia page for Project Pegasus says that:
Project P.E.G.A.S.U.S. (or Pegasus) was the name of the joint
NASA-United States Air Force project run by Wendy Lawson. The goal of
the project was studying the Tesseract in order to build the
Light-Speed Engine. The Project was shut down upon Lawson's death in
1989 and restarted by the World Security Council and S.H.I.E.L.D.,
to continue experiments on the Tesseract, until the actions
perpetrated by Loki rendered the project defunct.