Almost nobody looks at the batch number. It is a short code stamped or printed somewhere unglamorous, usually the base of the jar or the crimp of a tube, and it is easy to read as a printing artefact rather than information. It is one of the more useful things on the packaging.
The code is usually readable once you know the shape of it. Most follow some version of the same pattern: month, year, a product code, and a sequence number for the run. So a code might tell you that this jar came from the third run of a particular product, filled in a particular month of a particular year. Formats vary between manufacturers, and there is no single standard, but the components are generally those four.
What it does is more important than what it says. That number is the link between the jar in your hand and the records held at the facility that filled it: which deliveries of which raw materials went in, which equipment was used, who was working, what the checks showed at the time. Without it, a finished product is anonymous. With it, a specific jar can be followed backwards through every step that produced it.
That is also the mechanism behind a recall. If a problem turns up months after filling, the batch number is what decides whether the response is narrow or total. A brand that can trace a batch pulls the affected run. A brand that cannot trace a batch has to pull everything, or worse, cannot work out what is affected at all. The difference is entirely down to record keeping that happened long before anyone knew there was a problem.
There is a simple test you can apply as a buyer. Ask a brand what their batch number means. The answer should be immediate and boring, because for anyone who has looked at their own records it is a matter of fact. A brand that cannot tell you what the code means is unlikely to be able to follow it anywhere, which means the traceability exists on paper and not in practice.
It is worth saying that this is unglamorous work with no marketing value. Nobody buys a product because of its batch coding. It only becomes visible on the day something goes wrong, which is exactly why it has to be in place long before that day.
Every batch we receive comes with a certificate of analysis, and we keep them.