Ok I'll bite. This is a 200+ long thread and no one has got this right.
What you suggest is actually illegal in the US since the 1980s by federal law and before that it was customary for us states not to track that information. On the public side it is illegal because requiring identification for almost any kind of state service must hinge on there being a specific reason inherent in the public service why the state government needs that information to provide a service. Only the federal government can require the kind of information that would commonly fall under what is required in an EU country.
Private businesses are actually indirectly forbidden by federal law from asking for any kind of information like this. If in the course of asking for such information the private business discovers or deduces the 'immigration status' of the person that business now has to explain the reason why it discovered that fact and the burden mostly falls on the business almost every conceivable case. Therefore, most businesses are certain never to ask about such information. The only information transmitted to the federal government (by federal law) is essentially a self-certified form filled out by an employee only in the course of seeking employment. The business is directly forbidden by law from asking anything about the immigration status of the person. The state can't do anything about this or indirectly require a private entity to act based on such information.
One can quickly see that there is no workable way under current federal law to make any kind of system you would see in the EU or most of the western world workable. You couldn't directly collect useful information about a person in almost any way and any information you could collect wouldn't be useful.
Here is the politically incorrect fact: It makes immigration enforcement almost impossible if you can't restrict access to private employment or public services. It stands to reason that lacking such enforcement would induce large amounts of illegal immigration.
This isn't really a side effect either. The political economy of federal immigration law has something to benefit both parties. Conservative Republicans disproportionately benefit from cheap(er) labor and democrats/libertarians get to reduce the power of the federal government in ways that they see as politically desirable. There is a reason most immigration laws until the 2000s were overwhelmingly bipartisan.
This doesn't have anything to do with voter suppression at all. Any plan to require or give out ids for voting runs into the issue of how to handle these legal issues. One solution is for the government to only require the id for voting and nothing else at all. In other words, the government will get to have access to personal information but: 1) can not make using the id service convenient(as covered above) and 2) cannot use that information for any other public reason(unless federal law is changed) finally, 3) only the federal government can make such information useful which means you rapidly start approaching a centralized database which most european /western countries do not need to have.