Fixed-width serial numbers are just easier to handle. It also allows you to know where the serial number is within a overall product ID string without special rules to parse the string.
For example, if the serial number is expressed in hexadecimal in the string, it can contain letters A-F in addition to the digits 0-9. If the part ID string might also contain a suffix or prefix with these letters, then it becomes ambiguous with a variable-length serial number and without some delimiter. Consider "07933A". Is the whole thing the HEX serial number, or is 07933 the serial number and "A" a suffix denoting a configuration option, for example.
Fixed-width serial numbers strings are also easier to see in a list, and sort on.
Serial number generation
You didn't ask about this, but you have to think carefully about what/how new serial numbers are generated. This is probably more important than how exactly the serial number will be listed on the nameplate. I have seen a number of screwups, usually as a result of giving manufacturing too much leeway, or too much information so that they think they know how to create new serial numbers.
What I usually advise now is a dedicated piece of hardware that is pre-programmed with a range of serial numbers, then gives out those serial numbers sequentially. When the range is used up, it refuses to give out another serial number. The hardware must be returned to a central serial number management place to get updated with a new range, or swapped out entirely. You don't give any documentation about the internals of this device to manufacturing, whether internal or contract, remote or local. This device is a necessary part of the overall production process, usually part of the final test jig.
One advantage of this method is that it's easy to assign serial number ranges to different production lines or even different manufacturers. Engineering will need to create internal prototypes and sometimes samples, so they have one or more of these units too. Any engineer can grab the unit floating around engineering to assign a guaranteed unique serial number to a new prototype.
This dedicated hardware need not be big or expensive. It can be as simple as a USB dongle or small device connected to the USB. I've done this with a USB microcontroller and external EEPROM. Again, production gets no documentation on this device other than how to use it. They can be given multiple devices as spares, which is OK since each has its own unique serial number range. We usually programmed these devices with a range of 65536 (216) serial numbers. To update the serial number range required soldering a temporary jumper between two pads for that purpose, and running a special program on the host that only one person at the home office had.
Screwup #1
Apollo Computer made workstations in the 1980s. Each had to have a unique serial number. Among other things, these serial numbers were used to allow each node to make its own globally unique ID numbers used in the file system. The serial numbers were a hexadecimal string. A bunch of years after production started, someone noticed that the serial number of their node was well beyond the number of nodes supposedly produced. It turns out production was assigning serial numbers in decimal, even though each digit was really hexadecimal. This was fixed by having them make new units by filling in the unused serial numbers with digits A-F in them.
In this case no real harm was done, except that it made tracking units by serial number ranges awkward. Below a certain number, ranges were divided into decimal-only and hexadecimal.
Screwup #2
Serial numbers were assigned by the production test system, which included a PC. There were multiple production lines, and each PC was carefully assigned a particular range. The contract manufacturer in China was given explicit instructions to notify us in case of any trouble with any of the production test systems.
One day someone noticed that some of the 10s of thousands of little gizmos received from the factory in China had duplicate serial numbers. This was bad, because the larger system these gizmos were part of relied on the serial number to uniquely identify each gizmo, which was in large part the point of the system.
It turns out that the factory in China had done something stupid with one of the production test PCs and messed up the data on it. Instead of telling us like they were supposed to, they knew they could fix the problem on their own and we'd never know. Their fix was to do a complete disk copy from the working system to the broken system. From then on the two systems assigned duplicate serial numbers.