Kinda yes, kinda no. At the very least a caliber can be determined with some accuracy (like, you can tell it's at least a .30 cal bullet but may not know if that came from a 300blk or .308 since they use the same bullet with different casings/powder charges, then however you may be able to deduce further based on the wound patterns but I'm not an expert really.

Matching the bullet to the specific gun down to the serial number is a bit of nonsense though, to some degree the striations may match up like on TV but that'd be able to be hidden by taking a wire brush down the barrel or just shooting a bunch more rounds through it. That said, idk, maybe since the story is "one bullet fired, rifle found in woods, next bullet was ballistics testing," maybe it's more credible?

In any case, there's more questions than just the bullet "matching" (whatever that would mean in this context, be it striations or caliber), supposedly he climbed on the roof, assembled the rifle, took his shot, and then the video footage of him running across the roof. But in that footage he has no rifle, he runs, tosses his bag, follows it down, and keeps running. Where did the rifle go? Did he take it back down and stash it in the ~6sec it took between the shot and him moving? Doubtful. And supposedly he left his screwdriver up there, how then is it taken down in the bag in the footage then found assembled in a box in the woods if the screwdriver he needed to use for that was still on the roof? And supposedly it was a 30-06 rifle, a round powerful enough to go through an elk at 2x+ the (iirc) ~150yd distance, and Charlie caught it in one of the softest parts in the human body "because he's superman actually." Fucking ok, for sure, you betcha.