The president has no authority to do this. Federal departments and agencies are named by Congress, and even the Republicans in Congress have shown no interest in formalizing this.
Sure, no such law that I know of. But there's also no law that suggests that anybody else needs to refer to the Department of Defense using terms that the president and his minions just made up out of thin air. I'm also arguing that going along with them, by itself, is harmful to a democratic government.