Currently, our fiat transactions, aside from physical cash-to-hand, they are "kind-of" safeguarded by the centralized banking system, in-terms of some level of anonymity. You can't walk into the bank and ask the teller to show you your neighbor's financial transactions of last week. In the de-centralized dream many of us hope for, there will be no central authority to keep some necessary level of privacy. Anonymity must be built-in to the currency.
ehm this was not the point of bitcoin anonimity, the anonimity that everyone want is the one that prevent the centralized autorithy to know about your movements not a random dude that can't do shit about it
so no any fiat is not anonimous ever, if taxman really want to investigate they can even know how many cent you are using to purchase useless things, everything that travels on big circuit like visa and mastercaard is 100% traceable by banks if they want
in this sense bitcoin is much more anonymous than any fiat, and do not forget that with bitcoin you can not actually connect any address to a identified person, i mean the blockchain does not tells at who those specific address belongs