cant you get in trouble if its serial number is not attached to your name?
Not if you buy the gun from a private party.
It's actually illegal for the government to 'track' standard firearms. As this presents an infringement on your Second Amendment rights to 'bear arms'. The anti-gun people continually try to make more laws and permits to get around this, but thanks to the Supreme Court, it keeps getting thrown back. It's an ebb and flow deal for sure.
The only weapons the Federal or State governments are allowed to 'track' are those weapons that qualify as 'machine guns' and they 'track' those through the use of the $200 tax stamp that you must purchase when owning this type of gun. Additionally, when you purchase the Tax Stamp, you agree to allow the FBI, ATF, or any other qualified law enforcement access to your homestead at any time to 'verify' the location of said weapon.
You can get into big trouble by having a gun without a serial # (unless it is verified as an antique). You can get into big trouble for removing the serial # from a gun as well.
But as far as 'registering' it, a dealer will know all the ins and outs, do's and don'ts.
Shotguns are very minor in the world of guns and 'tracking'.