The fed is organized a bit differently than other government organizations, though similarly to how the US Postal Service is organized. But those are mostly to give it some semblance of independence from the winds of politics. But it is very much under the control of the US Government about as much as the US Military is under the control of the US Government.
From the fed's own website (notably a .gov domain):
"Some observers mistakenly consider the Federal Reserve to be a private entity because the Reserve Banks are organized similarly to private corporations. For instance, each of the 12 Reserve Banks operates within its own particular geographic area, or District, of the United States, and each is separately incorporated and has its own board of directors. Commercial banks that are members of the Federal Reserve System hold stock in their District's Reserve Bank. However, owning Reserve Bank stock is quite different from owning stock in a private company. The Reserve Banks are not operated for profit, and ownership of a certain amount of stock is, by law, a condition of membership in the System. In fact, the Reserve Banks are required by law to transfer net earnings to the U.S. Treasury, after providing for all necessary expenses of the Reserve Banks, legally required dividend payments, and maintaining a limited balance in a surplus fund."
https://www.federalreserve.gov/faqs/about_14986.htm