A social security card is not identification. It has no photo on it. Yet, as others here are saying, many organizations use the number itself as identification when it was never intended for that purpose.
Perhaps the answer lies in the fact that the US has no national ID. IDs are issued at the state level and the details vary from state to state. This helps make it easier to forge identification because locals will not be familiar with the standards of all fifty states. So, due to the lack of a national ID, we use the federal social security number in a manner it was never intended to try to fill in this hole in our system.
The political organization of the US is a little bit like the EU rather than a country per se. Each state is a little bit like a separate country in some ways. There are overarching federal laws, but a lot varies from state to state, just like a lot varies from country to country in the EU.
Perhaps the answer lies in the fact that the US has no national ID. IDs are issued at the state level and the details vary from state to state. This helps make it easier to forge identification because locals will not be familiar with the standards of all fifty states. So, due to the lack of a national ID, we use the federal social security number in a manner it was never intended to try to fill in this hole in our system.
The political organization of the US is a little bit like the EU rather than a country per se. Each state is a little bit like a separate country in some ways. There are overarching federal laws, but a lot varies from state to state, just like a lot varies from country to country in the EU.