Most firearms are not registered.
A popular CSI is based in Florida, where registration is not only non-existent, it is against the law.
As for fingerprinting:
'Ballistic fingerprinting' is greatly exaggerated.
It make tell make and model, but the rifling on a specific firearm changes through the course of use and wear.
It may give one 'fingerprint' one month, and quite a different one many shots and cleanings later.
This contrasts with real fingerprints on a hand, which regrow with the same pattern each time. So the terminology is misleading.
In fact over the course of its lifetime firearms of the same make and model can go from different 'fingerprints', overlap at some point, and then have very different fingerprints later on.
This means several firearms in the United States may match the 'ballistic fingerprint' of a firearm used in a murder, even when the actual firearm used may no longer match. Your firearm may match and cease to match several other firearms of the same make and model throughout its lifetime and wear.
Recovered bullets also are often damaged, and some firearm models have such close rifling between individual guns that telling them apart to within a specific firearm after the bullet has been deformed is unrealistically performed in tv shows.
Ballistic fingerprinting only has much potential when limited to the small number of firearms used in crimes. When you expand the database to all firearms there is so many potential matches that it can hinder more than help.
It can also lead to false leads, suspecting innocent people because they have a certain make and model firearm. If the firearm owner actually lives near or has had some form of contact with the victim it could even lead to false convictions, with investigators building a case against innocent people when they lack leads in another direction.
For example someone is killed with X make and model firearm. They look in a database and see 10 people own that make and model firearm. One or two of them is a partial match to the deformed bullet. The actual killer likely has a stolen firearm which may be from far away.
Most criminal guns are stolen. This means whoever they are registered to when registered is often not the murderer.
This means a database rarely does any good unless the firearm has already been recovered from the possession of the person that misused it.
If it is still in the possession of the criminal tracing it to the lawful owner that had it stolen does no good.
If it is stolen it typically has no official links to the individual that misused it.
Drug addicts often steal and rob and sell stolen property to fund their addictions.
The gang members or dealers they obtain their drugs from may be offered stolen items, like firearms. The gang members often share or sell some of the firearms they obtain to other criminals.
A gun can go through multiple criminal hands before it is recovered, and so even the criminal found in possession of it may not be the one that actually committed a crime with it.
In all registration is less common than in CSI, ballistic fingerprinting is greatly exaggerated on tv, and registration and fingerprinting poses a major threat of confiscation or imposition of new restrictions on known gun owners when politicians know where all the legal firearms are.
It provides few realistic benefits and many risks to gun owners.
TV is fantasy, but that fantasy has a real impact on reality. Many potential jurors for example think CSI is real, and are unrealistically persuaded for conviction by 'physical evidence' when it is presented and led to conclusions the physical evidence may not really prove, and unrealistically biased against conviction when that physical evidence is not presented.
CSI has helped to ruin the jury system by creating certain expectations. Expectations that if not met they don't wish to convict over, and if met they are overly eager to convict because of.