It's not "The ICA", there's a different agreement for each company. Some might not even prohibit you from revealing client information.
But as a rule, treat propriety information as confidential. Technically you shouldn't even tell your family who shops who. In reality, this is silly because most employees know who shop them. It's something that routinely gets leaked around the workplace.
And yes, you better believe employees from various companies sign up as shoppers. I was talking to my son's friend (who works at McDonalds), and he told me that his coworker not only knows the correct company that shops them, but he signed up for that company and has the questionnaire.
If I can personally find an instance of this happening, you better believe this is happening regularly. Especially among companies that offer rewards for successful shops.