Depends on the type of gun & state. Even in ridiculous NY, you don't need a permit to purchase long guns. You do, however, have to pass a background check. I presume that due to the background check, a record is kept of what gun you purchased.
I'm not sure about fringe, anti-American states like the ones that ban large sodas and pass 7-round magazine limits even though most guns don't have magazines that small so all the criminals who want to kill you are on the honor system when they load their stolen guns, but, most of the time, a background check itself is not a record of what gun you purchased. You get the background check before you purchase the gun. And maybe you get the background check but then changed your mind on purchasing anything.
But, gun dealers are required to keep a log book of sales, so the government could track the background check and then manually go to the store and check the log book to maybe put 2 and 2 together. That is gun registration in a way, even though gun registration is illegal. And then some people wonder why other people don't trust their government.
I think the Constitution was a great accomplishment in forward thinking, however the reality of when it was written vs the reality of now leaves just too much open to bias interpretation and abuse.
The Constitution doesn't grant rights anyway. The 2nd Amendment doesn't grant the right to bear arms. Those rights are inherent with being a free man.
Frankly we need a new constitution we can use similar ideals and values as it applies to the modern world.
The modern world has governments that have murdered 100 million of their own citizens, after first disarming them.
Are they really that concerned about the government getting out of control?
Consider this, only a few years ago a president could not get a wire tap without a judge. Now he can kill you.