No? Money simply has a signature. The major reason why cash is "anonymous" is because there is no attempt to track the serials beyond initial deliveries to financial institutions, etc. Money already is simply pseudonymous; if, for example, we tracked serial numbers during transactions, you'd be trackable unless you exchanged currency outside of a bill reader setup.

But this is true of electronic systems as well. I could just as easily give you an SD card with $1,000 on it and the tracking system would fail rapidly.