Apple Music is heavily integrated into your iPhone’s Music app. This is great if you’re a subscriber, but the constant prompts and reminders to join get pretty tiresome if you’re not. The good news is you can remove Apple Music from the Music app. This will make it much more like the old Music app.
Go to Settings > Music and turn off the "Show Apple Music" toggle.
This removes the "For You" and "Browse" tabs, replacing them with a "Connect" tab. It also stops the Music app prompting you to sign up every so often.
As a dedicated Spotify subscriber, I’ve no interest in signing up to Apple Music. It just isn’t right for me. I do, however, still use the Music app from time to time to play the songs I’ve bought through iTunes over the years. At least with Apple Music turned off, the Music app is usable for non-subscribers.