Same here.
I set up my new Ultra 2 watch and chose the restore from backup option, to get the settings from my previous Ultra watch. Apple Music never synced any music, whether one song, a playlist, or recently played. Just the spinning circle then nothing. Yes, I have the latest OS updates on both my iPhone and Watch. Yes, I have Bluetooth and WiFi on and connected with both. Yes, I have the watch on the charger. Yes, I have the latest Music app. I get notifications and content from other apps like iMessage.
Podcasts are also not syncing correctly, though better than Music which is completely broken. The sync state for Podcasts on the watch doesn't match what the phone watch app Podcasts state says.
I have unpaired the watch, reset it, and set it up again from scratch. It still doesn't work. This is incredibly frustrating. Outside of the Activity/Running Workout app, music + podcasts are the main things I use on the watch. They go together! If this doesn't get resolved, I'm returning the watch because it's useless.