The cash register has many fathers. The history is quite murky. (Ignore what you find on Google.) I look at the 1840s as the period when department stores became possible because of the invention of the cash register. Department stores began appearing everywhere in the commercial world.
The cash register was nothing more than a cash drawer which was operated along with a manual posting of the total sale price and the ring of a bell. This invention, in its many forms, allowed for the management of retail. Managers could oversee many sales.
Retail required that all prices were fixed and not negotiated. Retail management required that managers know what was being sold at a distance from the actual sale. Retail management is by definition the origins of department stores.
You can see the important role of cash registers in commerce. When mechanical cash registers arrived by the 1880s, they were immediately successful.