How do I sign in to my existing Apple Music account on a new phone?

Use the same Apple ID. That’s it. A different Apple ID wouldn’t be able to access the existing subscription.

Verify that Show Apple Music and iCloud Music Library are activated in Settings﹥Music﹥.

Access your music collection on all of your devices with Apple Music - Apple Support

You can go into settings and sign into the iTunes and App Store

also in the music app in the For you tab - in your profile (upper right corner) - scroll down to “ View Apple ID” - sign in

Back in settings- Music- turn on “iCloud music library” and Show Apple Music”