I believe it has to do with "public broadcast". Playing anything in private is one thing, doing so across large crowds or generally in the open is (often/usually) defined as a public broadcast which then, yes, enters the arena of copyright protection. Another angle is the use of copyright material in advertising, and a copyrighted recording (and much else) being formed into apolitical slogan is again a copyright matter. The concept of intellectual property (one on which the USA postures as a "champion of rights") alone, copyrighted or not, is likely also to be cited in a case like this. Even a private individual with no copyright, artistic product or intellectual property at stake could plausibly raise an objection to being used in an advertising/campaign sort of way - consider if a photograph of that person were to be used over and over again by Trump, Putin, Orbán, Erdoghan or others to symbolise what they and their supporters are opposed to (China, LGBT, foreigners, Kurds, somebody/anything). It is the act of ,contrary to the owner's stated wish/will, appropriating something belonging to someone in order to promote a position/view/agenda that is likely to place the one who takes the liberty in a position of guilt.