I can't recall if the serial number is shown when you go to the settings screen by holding the on-screen remote icon.
That unit has not been produced for 18 years and is no longer under warranty. Why does it really matter? Are you trying to match the unit to a box with a serial number? Usually the remotes, docks all have part numbers on them. When sold as a set, the box has another part number.
Maybe this will help decipher...[Link: tmraudio.com]