The UPS Store franchise began operating under its current brand name in the United States in 2003.
The business originated as Mail Boxes Etc., a packing, shipping, printing, and mailbox-services franchise. UPS acquired Mail Boxes Etc. in 2001 and began converting U.S. locations to The UPS Store brand in 2003.
The rebranding connected neighborhood shipping stores with the UPS delivery network while preserving services such as mailbox rentals, packing, document finishing, and private-carrier shipping. Not every location worldwide received the same name; many international Mail Boxes Etc. stores continued using that brand.
This question differs from asking when the first Mail Boxes Etc. stores adopted the name. The relevant date here is the start of The UPS Store branding in the United States, which was 2003.