> each legitimate bill has a unique serial number, and yes, at least at present I haven't heard of private actors outside of those that service law enforcement tracking it

This isn't quite what you meant, but 20-ish years ago there was some crowd-tracking of Euro-bills.

The Euro was pretty new and people were excited about their banknotes showing up in other countries. Some people created a website where you could enter the serial number of your bills. Each serial number would have a tracking page in which places it showed up. It felt kinda cool to find a bill in your wallet that had visited a few other countries.