Yes, well, sort of. The ballistics on the bullets would change, but the markings on the spent shell casings wouldn’t. These are things like extractor, firing pin, and ejector all leave unique markings on the shell. This isn’t quite as accurate as bullet striations, but replacing the barrel won’t affect them. To get rid of those you’d basically need to rebuild the gun from scratch each time.
Swapping out weapons would create a situation where, “well, people are getting shot.” Swapping out barrels creates a situation where, “well, there’s this guy who really likes P99s/USPs/whatevers out there shooting people.”
As an investigator, digging through looking for someone who’s chewing through guns like crazy, you’re more likely to find someone who’s just going through the barrels constantly.
With shotguns (loaded with shot shells), there’s no real forensics from the shot, but the shell casing is the forensics. In those cases, swapping out the barrel would be pointless. (There’s a longer discussion on the subject here.)
With high end precision rifles, you can swap out the barrel, but the weapon’s accuracy will suffer for it. If the marketing claims are to be believed, anyway. Which means if you’re character’s a sniper, that’s out.
That wouldn’t matter with a cheap, off the shelf hunting rifle, though, again, replacing it wouldn’t really be that much more expensive.
With automatic weapons, and semi-auto pistols, policing his brass is going to be basically impossible, so the spent shells would expose that it was still the same weapon, even if the barrel was swapped.
Revolvers step around the spent shells issue nicely, since they only eject shells when you empty the cylinder, but you can’t replace their barrels, at least not in most cases. So that’s out.
Also, with some heavier automatic weapons, including LMGs, they actually ship with replacement barrels, because you will overheat them during normal combat use. So this isn’t that strange a concept, really.
Once the ruse is exposed, tying the rounds back to the same shooter in court wouldn’t be that difficult. He’d actually be making the prosecution’s case easier, because forensics with shell casings aren’t as precise, but then he wouldn’t be able to challenge that, “no, the ballistics on the rounds themselves don’t match.”
It’s just safer for your hitman to dispose of used weapons and get a clean one for each job. He might carry a personal backup, that he only uses and replaces in an emergency. But otherwise keeping a weapon around would be a liability. It’s one more thing that ties him back to a corpse, if something goes wrong.