Your ssn isn't required to identify you for reporting purposes. You can be verified through data and association. While the card will be reported to the 3 main bureaus, there are other, smaller ones that are also used, some that excel at collecting random data. For example, I recently had to answer some verification questions for Discover. One of the questions was about my father and what age range he currently falls into. I've never mentioned my father to Discover. They got that information from a bureau.
There is a drawback to this and that is where your ssn would come in handy. Sometimes the data is wrong. Another example. When I was 18 or 19 and living with my parents, I applied for a Sears card and was denied because of a bankruptcy. I never filed, but my dad had. We have the same middle and last name, lived at the same address and both of us were employed by the Navy. The bank would not recon despite the wrong information and I was blacklisted for that card.
