The user who transfers the cash is anonymous if the user elides his/her identity in the process of that transfer. That the cash has the property of not automatically carrying the user's identity is incidental, because for example the cash can't turn off the surveillance cameras. You've conflated where the anonymous originates from, i.e. not from the cash.
In other words, anonymity -> fungibility not the other direction on the arrow.
Ah. Now I see where you're coming from.
If I may say so, that is the most ludicrous definition of "fungibility" that I've every heard. Turn out the lights and your money's more fungible ? LoL !
All I can say is - don't give up the day job. If Evan had lost any credibility in this thread I think he just got it all back

If you believe you won that debate, then you've lost credibility.
Fungibility is substitutability. It has nothing to do with anonymity directly and only is impacted when identity can be used to impact acceptability. Anonymity might aid the ability to be substitutable, but substitability doesn't do anything for anonymity.
Money is subsitutable because it all copies are the same w.r.t. to acceptability (e.g. serial numbers don't impact acceptability). That doesn't stop your fingerprints from being on the money. That doesn't stop the counter party to the transaction seeing your face.
Cash is substitutable for one reason because users accept it regardless of whose fingerprints are on it.
It is the inability of the government to go monitor fingerprints on each transaction that makes anonymity orthogonal to cash.
You are highly confused.