You would have to show ID if a police officer pulled you over. It's literally how the previous commenter described. The EU has the schengen area, which is an open boarders agreement that works nearly identically to the open boarders agreement between US states.
The reason the social security card says not to use it for identification is beause it's really bad at it. But it is used as an ID number anyway. The US doesn't have a national ID system in theory, but in practice it does. Not having an official national ID number just makes it less secure and convenient.