Apple Music is available on all compatible Apple devices as well as on Windows and Android. However, there’s no dedicated app for Ubuntu or any other Linux distribution.
Previously, Apple used to use iTunes as the platform for accessing Apple Music. This changed when Apple decided to split up the iTunes behemoth into separate, more manageable apps.
Since iTunes or any of the Apple apps dedicated to media playback are not available on Ubuntu, it was not possible to listen to Apple Music on the Ubuntu desktop without using a 3rd party service like Musish. I made a short video a while back explaining how simple it was to use the web based Musish service to access your Apple Music subscription and listen to your music collection.
Since then, Apple made Apple Music accessible via a web browser. I tested it recently in Ubuntu 20.04 using Firefox and it work perfectly. It was a simple matter of browsing to https://music.apple.com and logging in with your Apple ID. Firefox will update itself to add the required components for DRM music playback and you should be good to go. If you get the "Content_Equivalent" error, then add https://music.apple.com to your popup exceptions list within Firefox and your music should then playback without issue.
Once in, you will have full access to you entire music library including playlists. You can add music, edit playlists and playback your music collection as if you were in the proper Apple Music app.
I really hope Apple will release a dedicated music app for Linux someday in the future but based on history, I highly doubt this will ever happen. Still, this method of playing Apple Music through a browser is more than useable and will keep me satisfied for some time to come.