Serial Numbers are assigned by the manufacturer (or assembler or repairer - if different) usually against a customer/user specified tag Number.
Serial numbers are important when referring any spare parts requirement with the manufacturer, while that tag number maintains visibility of the valve in the customers/users environment.
Any 3rd party service organisation should use both references for any work/record.
Serial number should normally appear on a nameplate and also hard stamped somewhere on the valve, like on the outlet flange periphery/rim or under where the nameplate is (in case the nameplate gets removed).
Strangely ASME XIII Does not specifically state a 'serial number' is required, but at least the year built or a code issued by the manufacturer. However, API-526 states a serial number is required on the nameplate.
In practice you'll find that all manufacturers will use some sort of serial number (sometimes referred to a shop number).