In your first case, imagine that you have a contract manufacturer that is told to build something according to a particular Bill of Materials. You change the firmware and assign it a new part number (or assume that the version is embedded in the part number). Internally, the BOM is updated with this new part# and as part of your process, the manufacturer is sent the new BOM. Manufacturer goes to build the product and discovers that the firmware they have is a different part number than on the BOM. If not for this, they'd be building with the wrong firmware version.

In your second case, if the only person loading it is the customer, a part number may not solve anything other than the business managing inventory. However, if you're already in the habit of assigning part numbers to everything you build (I have come to be a big advocate of this), then it really is just part of the process.

I've seen a mix of both: there is a standard firmware version for the hardware combined with a set of customer customizations. In this situation, not having a unique part number for each combination (of firmware + customer config) resulted in confusion, angry customers and a manufacturing department having no idea exactly what it was that they were supposed to be building.

Yes, there are other ways of solving these problems but assigning unique numbers works well enough.