The authenticity of goods has nothing to do with the name of the store. The industry and commerce will register XX store for you only if you have a power of attorney, and will not register you without written authorization.
If you don't use the word XX store in your store name, what you sell in it is your own business, but don't make it too big and attract people's attention. When the time comes, the industry and commerce will check you, and the manufacturers will also find you in trouble.
By the way, when I was searching for Chinese trademarks of Adidas, I found more than 25 categories 170 registered personal names. In addition, adidas are not allowed to enter France. I'm confused!