Tony had 5 years post Avengers: Infinity War to have mentioned that he made Peter an Avenger and considering how much it affected him I'm sure he did. It was even one of the first things he says upon returning to Earth. In fact he even says it before asking where Pepper is, remember Tony is a father figure to Peter and Peter is like the son Tony never had.
Tony Stark: I lost the kid.
It seems highly likely that he would have told the other Avengers and certainly would have told Pepper. Whilst he personally wouldn't have told Fury it seems like something someone else would have told him or he left a note along with the E.D.I.T.H. glasses when preparing them.
It's also worth noting that Fury may not have known that Peter was an "official" Avenger, only Tony's protegee. Fury could just be paraphrasing the situation to emotionally manipulate Peter to do what he wants. Also selecting Peter to have the E.D.I.T.H. glasses and essentially treating him as the new Iron Man is enough for Fury to connect the dots to believe Peter is an Avenger.
Lastly, in the post credits scene for Spider-Man: Far From Home it is revealed that,
that wasn't actually Fury but Talos. Talos is presumably largely ignorant to the customs of who is or isn't an Avenger and just believes any Earth superhero is one. It could just be one of those things that Talos would assume Fury would say but that he wouldn't to subtly imply it isn't actually Fury. There are a few of those dotted around the film.