Don't pretend that the government is without fault here. There's plenty of tax fraud that happens using social security numbers. People can steal your tax refund or even evade taxes by pushing them onto you through your social security number. The system is messed up, and the government is largely responsible for the mess. The Social Security Administration has to same security holes.
What you describe is ideal, but it's not what actually happened. The social security number has been used as identifier and proof of identification for a long time. Part of the problem is that it's from a time when technology did not allow anything more complicated. That's no longer an excuse though. Social security numbers should have been upgraded long ago.
That is a verbal amendment to your employment contract, and as long as you do not sign an additional contract that somehow says "No, this actually means this" feel free to fuck shit up in court. That is a breach of contract and the company will be liable for damages (if any) and at least be required to give up any assumed license.
Verbal employment contracts are enforceable in in all 50 states.
What you describe is ideal, but it's not what actually happened. The social security number has been used as identifier and proof of identification for a long time. Part of the problem is that it's from a time when technology did not allow anything more complicated. That's no longer an excuse though. Social security numbers should have been upgraded long ago.