The problem is that they are asking the wrong question. If the only thing you look at is whether the person who contacted you is using the name of a legitimate company, you are setting yourself up to be scammed.
The truth is that people who will steal thousands of dollars from you will also lie to you. Many scammers use the names of legitimate mystery shopping companies and even the Mystery Shopping Providers Association.
Never assume that someone who approaches you unsolicited and offers money is for real. Always check them out. And check them out properly. Here are the questions you should be asking.