Yes, and the second form of identification need say nothing about legal eligibility to work.
It would be a huge burden to ask the employer to verify these documents.
I have worked in and with several businesses that knowingly turn a blind eye to the fake documents they photocopy/scan and file. The system is working as designed, propping up the businesses profiting from illegal labor.
Those who claim to be upset about illegal immigration don't want E-Verify mandates. They want a permanent underpaid underclass.