I was able to fix the problem for my two Apple Watches (Series 3 and 4!). I didn't see my paired Apple Watches in Xcode 15.3. As I'm new to watchOS development, I don't know if the problem would already occur for earlier versions.
To make Xcode show my watches in the "Devices and Simulator" I did the following steps: (I don't know if all of them are really necessary)
disconnecting iPhone from the Mac (=no cable)
unpair the iPhone from Xcode
unpaired both watches from the iPhone (=basically erasing them)
disabling "Developer Mode" (in "Privacy & Security" settings) on the iPhone
execute "Clear trusted computers" from the iPhones "Developer" setting
restart the iPhone and Mac
pair the watch with the iPhone (this takes a while!)
make sure Xcode is not running on the Mac
connect the iPhone using a cable to the Mac
choose "Trust" when asked by your iPhone to "Trust This Computer?" and confirm by entering your PIN
go to the Finder, choose your iPhone in the sidebar and do the same thing: choose "Trust" to establish a connection
start Xcode and open the "Devices and Simulator" window, where you will see your iPhone and its paired Apple Watch.
again, you will have to establish "Trust" by confirming the connection on your iPhone and watch
finally, to be able to fully connect to your phone & watch, you will have to re-enable "Developer Mode" (in "Privacy & Security" settings) for both, the iPhone and the watch, and restart the devices.
now you should be able to fully access the devices in Xcode 15.3 even without a cable connected between your Mac and iPhone.
PS: If you have a Apple Watch Series 3 and 4 (as I do), you might perhaps struggle if the Mac, iPhone and watch are not connected to the same 2GHz WLAN... Make sure they are all connected to the same network.