The subtle way is to use the Watch app, My Watch, and in Music and Photos, turn off sync and with music also turn off downloading Recents.
The brute force way is to erase the Watch and re-pair as a "New Watch" but you will still need to turn off the sync.
Simple erase
Be warned theSeries 3 is a devil to pair to ios16
Press and hold the side button until you see the power button in the upper right corner of the display.
Press and hold the Digital Crown until you see Erase all content and settings.
Tap Reset, then tap Reset again to confirm.
Wait for the process to finish, then set up your Apple Watch again. Set up your Apple Watch – Apple Support (UK)
When asked, set up as NEW WATCH..
(Erasing all content and settings erases all media, data, and settings from your Apple Watch, but it does not remove Activation Lock if it has one.)