Hi, I'm John!
Just to clarify, are you trying to add the device on the Microsoft account devices page online?
If so, that page mainly recognizes Surface, Xbox, and some Microsoft-branded accessories. It often does not recognize serial numbers from third-party PC devices, which can cause the “serial number not recognized” message.
Also, for most Windows PCs, you don’t need to manually register the device. The device is usually added automatically to your Microsoft account once you sign in with your Microsoft account in Settings on the device. You can check this on your PC: Go to Settings. > Click Accounts. > Make sure you are signed in with your Microsoft account.
After that, the device should automatically appear on the Microsoft Devices page after some time.
Click "Add comment" to let me know.
Regards,
John J.D.