It's anonymous assuming the mint-chip value message and the mint-chip request message are not intercepted or stored. This is a PKI-based system, so they can verify if you signed a transaction. So if you use it online - you can't guarantee it is anonymous. If you use it in-person and nothing stores the request or response then, in theory, it should be anonymous.

The giant caveat is that the documentation online does not seem to indicate whether or not a transaction log is stored in the trusted hardware.