The behavior you describe suggests that the Mac is detecting the Apple Watch correctly, but the final authentication step is failing.
Since the Mac displays “Signing in with Apple Watch” and begins the process before stopping, the connection between the two devices appears to be working, but the authorization is not completing.
Because the feature worked before the macOS Tahoe 26.5.2 update, it is possible that the update introduced a compatibility issue or a software bug.
Before assuming that, it is worth resetting the Apple Watch sign-in feature.
Open System Settings on the Mac, go to the Touch ID and Password section, and turn off the option to use Apple Watch to unlock the Mac.
Wait briefly, then enable the option again and enter the Mac password if requested.
Confirm that Bluetooth and Wi-Fi are enabled on both devices, that both devices are signed in to the same Apple Account, that two-factor authentication is enabled, and that the Apple Watch is unlocked and being worn.
Restart both the Mac and the Apple Watch. For the Apple Watch, perform a full power off and restart rather than only restarting the Mac.
It is also important to confirm that the Apple Watch is running the latest compatible version of watchOS 26.5
Some macOS updates can affect Continuity features and require matching updates on connected devices.
If the problem continues after resetting the feature, and the issue began immediately after installing macOS Tahoe 26.5.2, it may be a software regression in that release.
Though my iWatch series 11 works just fine with a M4 Desktop with macOS 26.5.2