renewal of subscription

That's correct. It is a subscription service; you only have the rights to listen as long as you are subscribed. That isn't just Apple -- other music subscription services (and non-music such as Netflix) work the same way.

My personal opinion here -- if it wasn't like that I suspect virtually everyone would subscribe for one month, spend the entire month downloading thousands of things and then cancel.

With Apple Music you only retain the rights to listen for as long as you have a subscription.

If you purchased any music through the iTunes Store, you do not lose the rights to that music.