Yes, you need to prove you're employable. But it's possible to do so with documents containing no anti-counterfeit provisions.
I've argued with a guy on another board. He bitches about illegals. But he had a patio installed and went with a low bid. He "didn't find out until later" that the workers were illegal. I asked if he reported the business owner. Of course he didn't.
Here in Arizona, we had the infamous "Papers Please" law (SB1070) that had an employer sanctions provision. So did a previous law, since it was politically expedient to throw a bone to folks who thought employers were also breaking the law.
Wanna know how many business owners were fined under the laws? One. It was some small business in the middle of nowhere that got a slap-on-the-wrist fine. That was it. They never tried to enforce the provisions otherwise.
Here in MA I had to prove that I'm legally employable each time the company that I worked for got acquired. My daughter in the last few years has had to prove it 3 times at least. Easily done. A passport is easiest.
I have no idea how enforced the law is.
(And the US passport is easiest if you don't run into someone who can't comprehend you don't actually need two documents. My passport I actually use, my social security card I certainly haven't used since the 80s and I'm not sure I used it even then. Of course I can lay my hands on my passport faster.)