Here is my path to enable developer mode on the Watch, inspired by @tyxop's reply.
My specific issue: Xcode Version 15.3 did not show Apple Watch in Devices and Simulators. It only showed my iPhone 14 Pro, running iOS 17.4.1, but not my Apple Watch Series 9 with watchOS 10.4. This was on a MacBook Air M3 with macOS 14.4.1.
After trying many options listed in this thread, the watch finally showed up in Xcode. Still, I never reinstalled Xcode or unpaired Watch from Phone. Below is what I did and it might be related to the Watch finally showing up:
What did not work: I connected phone via USB, disconnected USB and then unpaired from XCode Devices and paired it again. It did so several times with restarts of Xcode and also switching Developer Mode on iPhone off and on, rebooting MacOS as well. Watch did not show up, nor did I get any trust-this-device query on the Watch, only on the phone. I also dis/enable bluetooth on the Phone, but again, this did not solve my issue. The Watch did not offer the Developer Mode setting in Privacy & Security, also rebooting the Watch in between had no effect.
Finally I created an SampleApp in Xcode with > new Project >Watch Companion App. As the Watch was not shown, I could only target the iPhone 14 Pro as device. As this was the first dev cycle with the iPhone, Xcode promoted to add it to the Developer account Device List and created automatically provisioning profile and execute that SampleApp on the Phone.
What worked: The SampleApp's Watch companion App was shown on the Watch itself (on the Phone Watch App, it was listed under the available apps section). When starting it on the Watch, a message was shown reading "app cannot be installed because its integrity could not be verified" (text my be a bit different, I did not reproduce it an noted it incompletely). I rebooted the Watch, which took quite long (Xcode still open).
When it came back, I could enable Developer Mode on the Watch (I do not remember Xcode's messages, but I think, the Watch was already shown in the Devices list by that time). After enabling Developer Mode on the Watch, I could target the Apple Watch for running the App and Xcode prompted to also add the Watch to the Developer Account Device List before the App finally executed on the Watch.