It's not an assumption, normally people DON'T keep track of any of this. When I go to the bar and buy a beer with $10, the bar tender is for sure NOT writing down the serial number on the bill, checking my ID, and forwarding that information off to the FBI. The $10 can sit for a day or two in the till, and then be returned to another customer as change. In this way it continues its anonymous passage through the economy.

What about coins then? Those don't have any serial numbers or anything like that, and there's some legal basis that makes that "untrackable" currency legal.