A trustee of a trust is the legal owner of the assets of the trust fund. However he his not free when it comes to decision making as he has to follow the objectives contained in the trust deed.
In the case of the Fed the objectives contained in trust deed are the objectives set by congress in the act that created it.
You may ask a private entity to act as agent for you, in which case again the private entity has only power to do what you ask it to do for you.
So when the Fed is saying that the institution is not a gov agency, the Fed is just stating that it is an independent entity from the gov of the day.
