Who is right? Well it isn't him.
If you buy a gun from a dealer you fill out a 4473 form. That form will have his info and it will list the guns purchased, but it is not submitted to the government. The FFL is required to keep it for 20 years in a filing cabinet somewhere, and if they go out of business they turn them over to ATF but ATF doesn't use those to build a database, they do store them.
Neither is it true every gun barrel is registered by the manufacturer. I'm not one but afaik only the serial number and who they sold it to is stored by the maker. When ATF does a "trace" they go to the maker and have them look up who the gun was originally sold to. If that goes to a dealer they then come to us and we then go through our records to find who bought it, etc. But there's no computer database tying guns to buyers and ATF is only authorized to do a trace as part of a criminal investigation. it's also a very infrequent thing.
Guns are almost all test fired by the maker, but not for registration purposes, for quality testing.
You'd be amazed how many people think there is already a federal registry of gun purchases. Many think it's the background check system, NICS, that builds it, but any record of a NICS search has to be deleted within 48 hours iirc, so 2 days after the background check is run there's no record it ever happened.
There is the paper 4473 record, and for some that's too much and while I think the risk is infinitesimal I won't blame them as the feds COULD theoretically change the law and come collect them all and build one, but it would take an act of Congress to do so. It's a risk, but there is currently no database being built and it would be a monumental task to get all those paper forms into a database.
There IS one "database" being built, and it's for multi-handgun purchases. If you buy more than 1 handgun from the same dealer within 5 days we have to submit to ATF a form listing those guns and your info. The idea is to try to stop straw buyers who are buying for the black market, but it's not like it's done any good. That form gets sent straight to ATF and they do look at those guns and check them in a database b/c we've been called back on them by ATF to verify serial numbers and such.
Edit: forgot to mention that in the 4 border states with Mexico they also do this for multiple long gun purchases. It's in court being fought by the NRA and others, but so far no luck. All part of the Fast and Furious attempt to justify more controls (esp. a registry) b/c of Mexican gun running.