All US companies must fill out form
I-9 Form (Employment Eligibility Verification) (
archive) for every person they hire in the US, citizen or not. US citizens, are of course, allowed to work here in the US. By default non-citizens are not, unless they come here on a work visa, or get some other official written authorization.
Per the instructions in the
form itself, the employee must offer proof of both their identity, and authorization to work in the US. The instructions section gives a list of type A documents which satisfy both requirements, or type B and C documents, which satisfy those requirements individually.
You might note that a US Passport is considered an acceptable type A document. As a Europoor, you might wonder why US citizens don't just use their passport instead of bothering with birth certificates and other paperwork. It seems to me that having a passport is much more common in Europe, almost expected. I don't know if international travel is more common in Europe, or if some countries just issue a passport by default to be a primary form of ID, or something else.
Most US citizens actually don't have passports. No passport or other documentation is required to move freely between the various member states of the US. The US itself is massive, taking up a major portion of the continent. Someone could travel millions of miles in their lifetime and never leave the US. International travel is more expensive than domestic travel. Passports have to be requested, and are not issued by default. So unless a US citizen is rich enough to go on vacations to foreign lands, or has a job that sends them over international borders, most US citizens will never need a passport, and so never get one.