Axe I/O Solo allows "direct monitoring," which means you can hear your guitar playing through Amplitube without any latency (delay) while using your headphones plugged into the Axe I/O Solo. Amplitube 5 Max doesn't directly integrate with streaming services like Apple Music.
Direct monitoring is just that - you'll hear the direct (pre-amped) signal from your guitar, bypassing the computer - ie. NOT through Amplitube.
Use a separate audio player application like QuickTime Player or VLC Media Player.
A separate audio player isn't necessary. Just set your system output to the AXE I/O, hit play in Apple Music and jam along.
While Apple Music uses Digital Rights Management (DRM) encoding, it shouldn't affect your ability to play the audio through Amplitube
You don't need to play Apple Music's audio through Amplitube if you just want to jam along. If you want to use the multi-track recorder in Amplitube, you'll need a way to route the Apple Music audio. If you're on the Mac I can recommend Blackhole or Loopback for this.