Social security numbers are most emphatically not identification. The social security department tried for many years to get people to stop using your social as a form of identification, but companies (and sometimes even the government) still keep doing it. Social security cards printed before 1972 actually had a disclaimer printed on them that said "Not for identification purposes", but the message has since been removed. Although rare, there are actually duplicate social security numbers as well, and not everyone has a social security number. A number of religious groups, most notably some sects of Amish, refuse to get social security numbers and regularly fight legal cases to prevent having a social security number as a requirement for access to government services (although obviously not social security).
Popular misconception here: That disclaimer meant possession of the card is not identification, since it doesn't have any verification like a photo. The number does represent identification, as far back as 1943.
The US seems to have this quite backwards. They should be identifiers NOT authentictors although it is always likely to be slightly messy as some people won't have one and others may end up with multiple but having multiple people with the same number sounds pretty wrong.
It could be solved as pretty much every other country on the planet does it, with a national ID number (& card). But the US population rises up in outrage every time this is suggested.
Social Security numbers are not ID numbers, as other posters have written.
Other countries (including Israel, where I live) do have ID numbers, and they are used universally -- by governments, employers, credit-card companies, and the like. I actually very much like the convenience that this gives me. I have one number to remember, and I don't worry very much about it being taken or used by someone else.
Furthermore, many sources of credit information do not have access to your social security number to use as a key. Your phone company, cable company, oil company, and so on all may need to send your bills to collections and put a mark on your credit report, but they don't have access to your social security number (they may ask for it, but you aren't required to provide it).
There are cases of them being re-issued accidentally, but they're supposed to be unique, and we have massive data systems based on that premise. Texas has only just recently begun transitioning from using SSN as the primary key for their statewide student (and school staff) database.
when one credit agency merged me with another person, they put both our social security numbers on the credit report. I found it the weirdest thing in the world. They also combined our address records, so I had two different residences for my whole life, and residences for 15 years before I was born.