Label production and control is handled differently by a variety of companies, depending on volume and quantity of production and the processes involved.
If you stop to consider the difference between Boeing having hundreds of folks produce one aircraft and a small machine shop turning out thousands of parts per shift, you can get an idea of the range of possible schemes.
In my machine shop, for example, we generated inspection sheets, shipping documents, and labels simultaneously with generating the job traveler that followed the part from raw bar stock to packaged, labeled packages on the shipping dock. So the administration department which accepted the purchase order after Contract Review by a cross-functional team was responsible for pushing the button on the computer and starting the printing process. Our next door neighbor who ran 5-axis machining centers had his shipping clerk press the button on his computer to print the labels for his production. The guy who ran the grinding shop on the next block wrote his labels in longhand with black magic markers except when he was doing subcontract work for us - we gave him pre-printed bar-coded labels and special protective packaging.