Apple’s documentation confirms that the behaviour you described is expected.
After you turn on, restart, or log out of your Mac, the first login must be done with your Mac password.
Apple Watch Auto Unlock applies afterward when you wake the Mac. (Apple Support)
Therefore, if your Watch unlocks the Mac after you have manually logged in once, there is nothing wrong with Auto Unlock.
The fact that your Watch can also approve administrator-password requests is a separate Apple Watch feature. (Apple Support)
If Auto Unlock also fails after that initial password login, Apple recommends turning off the Apple Watch unlock setting, restarting the Mac, turning it back on, and confirming that the Watch is unlocked and close to the Mac. Apple also recommends current macOS and watchOS versions.
Apple’s relevant documentation:
Unlock your Mac with Apple Watch
Automatically unlock Apple devices — Apple Platform Security
Continuity features and requirements for Apple devices