Barrels are not serialized.
Bullets can be matched up to a point. What you're describing are known an individual characteristics, which are the unique properties of each barrel. Those properties change through regular wear and tear, or by deliberate action such as running a file or other abrasive down the barrel. My hunting rifle is a tack driver, but over time it will start getting worse and worse due to wear. In my case, it will happen quickly as the rifle is a magnum. If I were a suspect, the cops would need to get hold of my rifle quickly before I go for target practice because the properties of my barrel would change. In my case, I'd go to the range and fire off a munch rounds in quick succession, heating up the barrel so it wears faster. My rifle is uncommon enough that just based on the class characteristics, it may be enough to stand up in court.
On certain rifles, such as anything from Savage or an AR, it's pretty easy to change out a barrel, or even change the caliber entirely. I can change out a Savage barrel in about 20 minutes.
Of course, this is assuming that you have a suspect and a rifle to test.