There is a certain minimum amount of information legally and/or logistically necessary to become a registered shopper. Information such as your name, address and contact information - as well as determining that you are at least 18 years of age, are all critical.
Additionally, since most Mystery Shopping Providers (MSPs) remit payment via PayPal, they will need that information in order to pay you promptly.
Beyond that, there is a lot of optional information (that you do not have to provide) that makes it a lot easier to ensure you are offered shops best suited to you specifically - things such as whether or not you have pets, your gender, and the number of children in your household are good examples of this. Some shops require, say, a visit to a Veterinarian (and it's best to offer those to people with pets), or others may require visiting a Maternity Store to try on maternity clothes (which may require a female shopper).
Additionally, MSPs are required by the IRS to properly document all Independent Contractors - which is typically accomplished by providing a complete W-9.