Neither. You simply are not understanding what you are actually seeing.
The Android Music App, allows access to the Apple Music subscription service. If you don't already subscribe to it, paying for it, won't let you access your music on it at all. The Subscription service offers access to Apple's extensive music catalog for a monthly fee. Much like any other subscription service, such as Spotify, Amazon Music, Youtube Music etc.... No more cash grabs than those services.
If you already subscribe to it, and your music exists on a computer, you can use the iCloud Music Library option, and access your library in the Android Music App. But its just one option.
You can transfer music you own from iTunes or the Music App on a computer to your Android device directly by simply dragging and dropping the files onto the Android device however it is set to do so.
If your music is only available on the iPod, then you will need to extract it from there first.