(Organizations like the Red Cross are called “Congressionally Chartered Organizations” or “Title 36 Corporations”. See:
http://www.llsdc.org/sourcebook/docs/CRS-RL30340.pdf)
A key to the status of all these organizations is simple: in whose name do they file lawsuits and who represents them in court? The Federal Reserve files suit in its own name and defends itself with its own attorneys, not ones in the employ of the federal government.
The court case I cited was decided over the issue of whether the Federal Reserve was an “agency” of the federal government for purposes of someone filing suit against it. The judge ruled it was not an agency of the federal government.
OK, I accept that, but I wouldn't say that puts the Fed in the same category as private corporations. Or even in the same category as the Red Cross. The government still appoints and approves the majority of the Fed governors, and still maintains strict oversight with frequent congressional testimony.