"(above 16, and only qualified drivers, natch)"
No, you don't need to be a qualified driver, actually. Well, at least in Virginia you don't (in this schizophrenic country, who knows what other states do). You can go to the Dept. Motor Vehicles and get a picture ID even if you don't drive. Many students who came from abroad to stay a short period (like 1 year) and don't drive do that so they don't have to carry the passport around.
And anyway, the Social Security number thing ends up being the de facto national ID here in the US, they ask for that number for any stupid thing -- and that's when your name, number, address, etc. gets thrown into the wind, because they keep their data in some Win 98 computer running who knows what and connected to the internets 24/7 (discounting the few daily reboots).
"when you apply for your ID card, how exactly will you prove to the authorities that you are you?"
In Brazil, we use the birth certificate in order to then get the national ID number later in life. Do you have birth certificates wherever you live? Maybe you're not as organized, but we are supposed to register the baby as soon as it is born.
Now it's kind of funny, all the brouhaha about the national ID card. I don't know how many countries have it, but mine does, as I said above. Sure it's not the panacea to all humanities woes, but it's not like this Orwellian craziness you guys seem to think. Maybe your society have got so developed that you don't have any serious problems left to solve, so you go try finding something to cry about, like yet another number the gov will use to identify you (as if they didn't already have a bunch). Get a life, folks, or at least some sense of perspective...