It depends. Some MSCs provide reports with shopper names on them. Other companies keep the name of the shopper confidential. Still other companies do it both ways, depending on what the client wants. Some clients want the name of the shopper. And, if you make a purchase, with the receipt provided as a part of the report, the company can easily identify you unless you have paid cash. Even then, you may be viewed on video.
One client I specifically know about is Five Guys. I have not shopped them in several years because one of my friends is a FG Regional Manager. He receives reports for all FGs in his region, and the reports do contain the shopper name. He watches the video taken during the shop time, matching the reported details to the video. He does not provide the shopper info to the store managers or to any of the employees, but it is conceivable to me that his personal assistant and anyone in Human Resources may also have access to shopper names and may even have viewed the videos. I mystery shopped Five Guys several years ago, before he worked there, and he knows that I was a past mystery shopper, and we've discussed it.