Ownership is something ethnomusicologists deal with all the time, and the variety of answers to "who owns this song?" are much more varied than anybody's mentioned in this thread. Ownership can be individual or collective, with all sorts of collectivities and legal frameworks. Most of the traditional songs of one region pf Melanesia are owned by specific families, and you would probably be on the pointy end of a spear if you infringed that. One Native American culture expects a man to make one flute tune for himself in adolescence, played on a flute he made himself, which will be his exclusively, and he will never play anything else. Other songs are mysteries, the property of religious groups who never let outsiders hear them (like Aboriginal corroborree songs and the hunt songs of northern England). Others are taught and performed under compulsion, like national anthems, and the coercion is directed at those who don't join in.

In many of these situations the question "is it folk?" Is irrelevant, and if you really did want an answer, the status of the song (tune, dance, costume, ritual) in statutory or customary law wouldn't help you find one.