You cannot register your device on your account using a serial number, that option is only available for Microsoft manufactured devices like a Surface laptop.
For all other devices, ensure you log into Windows with your Microsoft account, then in the settings ap, ensure location Services is enabled for all apps and also enable the 'Find my Device' setting, your device will then automatically be added to your account online, that will not happen immediately, but it will happen.
There is currently a widely reported problem with devices showing up on the Microsoft account, it is a server side issue, but your device will show up on your account after some time.