The question of anonymity rises only with respect to online transactions. Off-line, anyone can offer a money holder $100 in cash, and receive the digital or other form of money valued by exchange to $100. The new owner of the money is of unknown identity to the seller, the government, and to whomever.
One who buys crypto money with owner-identified money provides a tie-in between his identity and the crypto bill, no anonymity.
Anonymity creates insecurity: no recourse in case of loss, theft, or robbery
Anonymity helps the law breaker, but also allows the law enforcer to entrap the law breaker.
Anonymity based on asymmetric cryptography (e.g. Bitcoin) is vulnerable to theft, or cryptanalysis of the secret private key -- resulting in total loss of the account, with no recourse. It is also vulnerable to court orders forcing a trader to surrender his private key.
David Chaum, 1990, pioneered a crypto solution that allows a buyer of crypto money to remain anonymous.
Chaum's solution is based on probability reasoning over blind signatures: the bank agrees to honor a digital claimcheck that is signed by the bank as bona fide, without the bank having seen what it signs. The bank exercises a crypto protocol that reasonably assures it that the blind signature is not a fraud.