Only the artists and their media companies "own" the music, you are licensed to listen to it. If this is music you have purchased from the iTunes Store you can retain and listen to it as long as you have the files. If it is from an Apple Music subscription then yes, you are renting it and when you no longer rent it then it will stop playing.
Apple does provide a re-downloading feature but does not guarantee a track will always be available for re-download. As long as you have backups then you don't need to be concerned if an item is pulled from the service and no longer available for redownload.
: http://www.apple.com/legal/internet-services/itunes/us/terms.html - "It is your responsibility not to lose, destroy, or damage Content once downloaded. We encourage you to back up your Content regularly." ... "Content may not be available for Redownload if that Content is no longer offered on our Services." And: https://support.apple.com/ht204930 - "Apple Music is not a back up service, so make sure to always have a back up of your music collection before you make any changes."