FDIC is an agency established and granted powers under US federal law, and its chair is nominated by the President, subject to Senate confirmation. Fundamentally, it is just as much part of the US federal government as any other independent federal agency, such as the CIA or EPA or NASA, is. The fact that, unlike most other federal agencies, it gets its funding through hypothecated taxation rather than general revenue, doesn't make it any less of a federal agency.

Also, while the law may call them "fees", they are taxes – it is not like banks have a choice in paying them – just because the law doesn't call something a "tax", doesn't make it not a tax (see National Federation of Independent Business v. Sebelius)