Music Rights generally refers to the the copyrights on music. It refers to the composers and performers. If the music is being used in public, such as department stores or restaurants, the rights holders are dues some sort of royalty. Many places use a "muzak" instrumental version of a song to avoid having to pay additional rights to the original perfomers. This is why some old TV shows have not been released on DVD or Blu-ray. If the show features recordings of popular songs, the rights have to be re-negotiated with the rights holders, adding to the expense of the release. In some cases, they replace the original music rather than pay the fees for the rights.
IMHO they should then be called Property Rights groups because 1) that’s the founding principal from with that derives and 2) it would drive the libs in the music industry crazy to have their group referred to that way!