We have something like this in Belgium with our eID cards. AFAIK the chip has a public/private key. I use a PIN code to unlock the private key and I can create signatures with it to prove my identity.

Websites can actually use this for proper citizen authentication (and non-repudiation). The integration has UX bugs, but at the end of the day, I am very happy that we have something much more secure than a US SSN.