Actually, they're not. They're a paramilitary organisation with members who happened to have ties to the US Government. It'd be like the BoS is THE US Army since their origins are tied to the US Army.
All evidence and dialogue suggests that the Enclave is either the actual US government or a presidentially authorized shadow government, thus they are legal. Their claim over the mainland may be subjective but they are the most direct successor to the US government you can find. The chain of command ends in them.
Maxon defected from the US army before the war and thus we can't say what became the Brotherhood is a part of the once-was US army. Had they stuck to the US army and maintained its traditions they could be called "the US army" or at least a part of it. The Enclave, on the other hand, definitely existed in pre-war America, controlled Poseidon Oil, had considerable state funding and maintained US governmental structure and its traditions.