The Fed is a private institution, though a small stake is owned by the US government. So it's not really one part of the state lending it to another part of the state.
"It is not 'owned' by anyone and is 'not a private, profit-making institution'. Instead, it is an independent entity within the government," however for our purposes this quote is probably more relevant, "the Federal Reserve must work within the framework of the overall objectives of economic and financial policy established by the government."
So, independent, but founded to enact government policy.