There is supposed to be a process called E-check I think. I'm not sure how it works as I'm not an employer. I don't know how it would detect a false ID if all the numbers matched?
Yes, it's called E-Verify. It doesn't catch every instance of those ineligible for employment, especially when it's down during the .gov shutdown.
https://www.e-verify.gov/
E-Verify has resumed operations. Given that E-Verify was unavailable for over a month, we ask for your patience as we reinstate the service. See E-Verify Resumes Operation for more information. Thank you for your support of the E-Verify Program.