When I was growing up and learning to drive in early 1970s New York, tag could also refer to the license plate itself, as the tag number refers to the plate #, and the sticker needs to jibe with that.
Memories of My Life and Travels
New Jersey police pulled them over because they had "New York tags"; in other words, license plates from out of state.
Life Don't Have to End
In this case, "paper tags" refer to those temporary cardboard license plates used by the distributor until all paperwork has passed and approved by the Department of Motor Vehicles. "Out of town" tags means from another state.
license plate (US)
a plate or tag, usually of metal, bearing evidence of official registration and permission, as for the use of a motor vehicle.
a numbered or lettered metal tag displayed on a motor vehicle, indicating that the vehicle has been registered with the state or country and may be driven on public streets and highways
So I believe the song refers to obtaining the final component to make the car apparently street-legal: the license plates.
It's actually the punchline.
Back in those days, believe it or not, as long as you lived in a small town and were on good terms with the local fuzz, you get away with most any infraction on the road.
Just about the worst thing you could do while motoring about was to get lost driving through that small town with "out of state" tags.
The abcolutely worst thing you could do is get caught driving black in a nice car in the same town. We used to call it DWB.