The United States has a dual banking system in the sense that

the public may deposit money in either commercial banks or savings-and-loan associations.

banks offer both demand deposits and time deposits to savers.

banks are chartered by the federal government and by state governments.

banks both take in deposits and make loans.

banks are chartered by the federal government and by state governments

Explanation:

Banks in the US are heavily regulated, specially by federal government entities, but also state regulatory entities.

Federal agencies that regulate banks are: the Federal Reserve System (FED), the Federal Deposit Insurance Corporation (FDIC), Securities and exchange Commission (SEC), the Office of the Comptroller of the Currency (OCC), and the Federal Financial Institutions Examination Council (FFIEC). Credit unions are regulated by the National Credit Union Administration (NCUA).

Banks are also regulated by state agencies (I guess every state has its own regulatory agency) like the California Department of Financial Institutions or the Nevada Division of Financial Institutions.