No, you can have the latter and have ID's for all people, like in almost any other western country. But due to the polarizing political system, apparently it cannot be done in the US, even though it would be easy, if the GOP/DEMs would display some common sense once in a while and work for the people and not against each other.
Yes, but only if those IDs are provided without monetary or opportunity cost to all people, regardless of spatial, economic and cultural situation. Anything more would constitute a poll tax.