Your serial number will be on the back of the charging cradle.
Galaxy buds are accessories and not a device, therefore they are unable to be registered.
When you registered your Samsung Fold it is automatic that your Samsung ear buds are recognized by the system since both are bundled. If for example you have any issues with your buds in the future you will need to provide the IMEI number of the Samsung Fold for the system yto recognize the buds that comes with the Samsung fold. Hope this helps.
Are you trying to register the buds? They are considered an accessory so you just need to register your phone.
I was able to register using the Model number listed on the box (mine is galaxy Buds+
my model # is SM-R175NZBAXAR
i used the right ear bud SN to register.
Hope this helps.