The CFPB is part of the federal reserve and both of those rules also apply to the federal reserve. It makes sense that a bureau of the Fed would follow Fed rule

The CFPB is part of the federal reserve and both of those rules also apply to the federal reserve. It makes sense that a bureau of the Fed would follow Fed rules. The Fed is tasked with financial regulation, and did some of the work of the CFPB before it was created. By design the Fed is independent, including funding, from Congress. If you're curious why, there's lots of history around how and why a central bank was created... dating all the way bank to Andrew Hamilton.