Matt Damon is Jason Bourne. That's an established cinematic fact, and it's the character that Damon is best known for even taking into account all the other great movies he's made during his career. Damon is so synonymous with playing Bourne that the one time they tried making a 'Bourne' film without Damon in it, it was a miserable failure. It very nearly wasn't his franchise at all, though. The part very nearly went to Brad Pitt instead.
Pitt didn't have a specific objection to playing Bourne; the fact that he would go on to play a slightly similar character in 'The Spy Game' a few years later tells us he's a fan of the genre - it just seems that he had too much work on when the role was offered to him, and so he had to refuse.