My music app requires wifi or cellular data

If you have subscribed to Apple Music, then you are streaming music, and that is why it requires data. If you download some content to your device, then you will not need data to listen. Use Apple Music in the Music app - Apple Support

The issue is with iCloud. If you have iCloud turned on, your phone will only play purchased music when you're connected to iCloud (usually via wifi.) If you turn off iCloud and connect your phone to iTunes (via USB to your computer for instance), iTunes will download all selected music to your phone and you'll be able to play it without wifi. But again, if iCloud is on when you connect to iTunes, iCloud will override iTunes and allow access to your music via iCloud only.