Many central banks are named "Bank of <Country Name>", like Bank of England (for historical reasons not Bank of UK), Bank of Germany, Bank of France (Banque de France) or Bank of Spain (Banco de España). If we had a private "Bank of The United States", then yes, that would be confusing, but no official institutions are named "BlahBlah of America".

Also, the fact that the official central bank is so close at "The Reserve Bank of New Zealand" makes it more confusing for those unfamiliar.