But your SSN won't give them any of that information... They aren't the government, its not like applying for FAFSA, which is administered by the government, where they can actually find out you are lying to get need based aid... The College Board (company who administers the SAT) is a private company, they don't have access to any information that you don't voluntarily provide to them and there is nothing stopping you from lying about any of it, apart from your SSN.
And once again with millions, perhaps even tens of millions (don't many overseas students take it as well to attend US universities?) of people both applying to take the test and actually taking the test, many of them several times per year, how could they possibly verify any of the information given to them on any meaningful scale?
Hell look at the college admission scandal recently... In many cases they can't even verify that the actual people taking the tests are who they say they are...