“Nothing ships without its number”: what does that actually mean?

It is a statement about sequencing, not a slogan. A lot is not released for sale until it has a lot code and a record behind that code: identity result, purity result, the chromatogram as issued, and the certificate of analysis. That order is enforced in how lots are published, not left to judgment on a busy day.

In practice it means the code printed on your vial is not decorative. It resolves, on Verify a Lot, to the specific record for the specific batch your vial came from — not a generic product page, and not a number invented to look complete.