I agree ISO/IEC 17025 does not require a calibration lab to supply the serial numbers of standards used for the calibrations their customers have paid for. The requirement for this kind of traceability is listed within ISO 9001, TS 16949 and perhaps other industry standards.
In these standards the language does not specifically say "calibration labs shall supply serial numbers on calibration certificates for the equipment they calibrate/verify" because the responsibility is placed on the client company to ensure that traceability is available. Historically their best tool for the job is to have the number on the certificate, but if there is something else such as a database they can rely on they could exhibit that as evidence they have the traceability the standards ask for. However it's done, there's an expectation that traceability can be established so as to lend confidence to the measurement process. Boom, that's it.
In these standards the language does not specifically say "calibration labs shall supply serial numbers on calibration certificates for the equipment they calibrate/verify" because the responsibility is placed on the client company to ensure that traceability is available. Historically their best tool for the job is to have the number on the certificate, but if there is something else such as a database they can rely on they could exhibit that as evidence they have the traceability the standards ask for. However it's done, there's an expectation that traceability can be established so as to lend confidence to the measurement process. Boom, that's it.