I believe previously your music was synced to each device manually where you could play the music on each device independently. When you subscribed to Apple Music, you turned on Sync Library and that uploads all of your music on that device to Apple Music/iCloud, where it is then automatically synced on all your other devices and the music is actually streamed instead of playing locally.
Apple Music has 2 plans, an individual plan that will only allow you to stream from 1 device at a time and a Family Plan that allows up to 5 simultaneous streams. I assume this is the upgrade option you are given in the alert.
You may be able to turn off Sync Library on the devices where you want the music to play locally instead of streaming to maintain the individual plan to Apple Music. Your music will not sync to those devices that have Sync Library turned off and you will need to sync manually to those devices like you did before. Check out these 2 support articles about the difference in syncing the music: