As the article mentions, keeping an up-to-date and accurate stock register is very important to the company and the stockholders. I didn't know that they had a special number for each share of stock.
The company employees who are in charge of keeping track of stock information when stocks are bought and sold. They must be really busy when the stock market rises or falls suddenly and there are massive amounts of stock shares bought and sold.
With well-kept records, historians or other interested people can get a sense of the history of the company by looking at the stock register.