There are no Federal Laws nor any state laws that mandate that someone must obatin any kind of ID or carry one.
Driving is a different story, you must get an ID and carry it.
This will be the interesting part, there are ways to conduct records checks based on a name/DOB/POB that officers can check for people suspected of being illegal.
Not so long ago to cross the border all someone had to say was "I am American" and that was it, if they seemed honest, they got in...
Then birth certificates were enough... (eaisly forgable)
Now everyone has to have a passport...