Our system of laws is complicated but there is (generally) some underlying logic to their structure.  In the case of radio transmissions from an aircraft it is necessary to understand that while the FAA could be thought of as “owning” your aircraft, the radios you carry in it are “owned” by the Federal Communications Commission.   

Except for military and certain other government operations, within the USA and its possessions all devices that radiate RF (and some other equipment too) fall under the FCC’s legal control, making the FCC  the primary government arm legally empowered to specify both the technical characteristics required of your aircraft radio and how it is to be operated. 

The FAA can and does add their own operations-specific regulations where they see a need, as in the separate FCC and FAA regulations regarding cell-phone operations aboard aircraft. 

Regulations do evolve, but at a glacial pace.  Older pilots remember that for decades every radio-equipped aircraft had to have its own “radio license” and the pilot needed a radio operator license or permit.  Today only those of us who fly internationally need to have those.  This is the result of a major policy shift that in 1996 led to, among other things, the FCC’s elimination of formal licensing for radios used domestically in most private aircraft & small boats.  However, even though you may have no paper license, be assured your aircraft transmissions are still regulated by the FCC, and the FCC officially demands you identify yourself on initial call.  

The current regulation on identification of aircraft voice radio transmissions is actually a compromise between the needs (wants?) of the FAA and FCC.  The FAA wants to hear the type-of-aircraft information plus a specific ID, while the FCC only cares that a specific ID identifying who is transmitting is given. Years ago, technically you were supposed to use your full “N-number with the N” on initial call-ups because that was (still is) your “official” FCC call sign, "N" being a USA call sign identifier. Some where along the line (1980's?) the FCC acceded to FAA’s desire for alternative call signs that included the valuable type-of-aircraft data within the call and rewrote its reg. 

Admittedly it would be an extremely unlikely event to be dinged by an agency currently more interested in selling off the public’s radio spectrum than enforcing regulations. But the regulation does exist and could be brought into play if "they" find it useful.