It is a case of lost in translation. Chinese use "invoice" for "receipt" because they use the same official form for actual invoices, I think, as a way of cracking down on tax avoidance. China has one of the highest sales tax at the retail level, including VAT and various fees; so high in fact the actual rate is a state secret. You can imagine the pressure on retailers, particularly, small shop owners, to avoid paying it. The government therefore introduced a system of official invoices that everybody including all retailers must use, and from carbon copy of invoices tax collectors can calculate sales amount and thereby tax owed.

That is my understanding, anyway. I have been out of the country for so long, it must have changed a lot. Most of it must be electronic now, I imagine. I still remember a shop owner's audible complaints when I asked for an invoice around the time the system was initially introduced.