I was able to fix this issue on My Watch 6 Classic and S23 Ultra.
On the Galaxy Watch: Settings -> Modes, Scroll down and disable "Sync modes with phone"
Then try enabling "Sync modes with phone" on the phone and on the watch again.
Now the setting "sticks" for me on both the phone and watch and the sync behaves as it should.
Hope this helps someone.
I have a Modes Icon on the watch Quick Panel. When I press that I am presented with a Modes Menu containing Sleep Mode. I press Sleep Mode and that mode is activated on the watch and phone.