well, as far as I know, the "approved for road use" exhaust has a proper "certificate of fitness" where is written on which bike it can be fitted. this certified that "dimensions / weight / coneection " have been tested by manufacturer and approved by authority for those specific bike models.
so usually this "serial number" referring to certificate of omologation is stamped on the exhaust, however, the direct link with the bike is on the certificate
see here a template or
here a certificate
so if your are holding the certificate and the Dbkiller is on there is no problem.
however, since usually the aftermarket exhaust does not comply with environment pollution restriction, where emissions are checked (both random or periodically) in Italy for example all cars and bikes must inspected after the first 4 years since they receive "plate" and than every 2 years, you need to resume the bike to respect emission to pass the check