So first, as your statistics state, 29.4% is the rate in the state of Mississippi and 3% is the rate in New Jersey. The other 48 states, DC, and the 5 inhabited territories fall somewhere within that range. Driving laws in the U.S. are set by the states individually so there isn't going to be a database of insured cars available on a national scale because the national government doesn't get involved in traffic law enforcement. Most insured statuses are thus provided by a proof of insurance documents by the driver on traffic stops. And it does get enforced, however, the cops won't know if you're violating it until they stop you for a valid reason. And cops can't just stop you to check if you have insurance or a valid license.
As for the disparity between the two states listed, that's a matter of where those states are. New Jersey is a small state with the highest population density of any state in the union. The northern half is practically a suburb of New York City and the Southern Half is a Philadelphia suburb, both cities famous for, among other things, not being in New Jersey... and cops love to give tickets to out of state drivers (they tend to just pay the ticket because going to court in another state to fight it isn't worth it... so, since most of the residents are commuting out of state, they aren't going to give cops another reason). But on top of that, New Jersey benefits from the many mass transit options that come with being suburban NYC... so if they aren't insured or have a suspended license, they don't need to worry too much about driving anyway.
Mississippi on the other hand is a very rural state... despite its larger size than New Jersey, the population density is way less. And one of the things I've notice when Europeans ask about why Americans do certain things... is that they fail to realize that America is so damn huge. To give you an idea of how big the U.S. if you took the First Transcontinental Railroad and placed it's western terminus (Oakland, CA) in Portugal and ran the line east from there, the eastern terminus (Council Bluffs Iowa) would be somewhere in Asiatic Russia. And Iowa is famous for, among other things, being about halfway between the west coast and the east coast in terms of the country. And Americans like that space because it means we can live in places where no one can reasonably bother us without having to commit to doing it. So... losing the right to legally drive in Mississippi is way more devastating than if you lost it in New Jersey. There is no reliable mass transit in Mississippi because there are not enough masses to transit... If you need to go take care of some business in town, the car is the only way to do it (I lived on the east coast in one of the most heavily populated parts of the country... and I had to drive for 10 minutes to get to the nearest "convenience store" from the house I grew up in. The interior is way worse). So if you didn't pay your insurance or you got your license suspended, and you have to go into town... so long as the cops don't catch you, there's nothing to stop you from driving there.
To go on what @David S. said, the German who has no insurance or drive on a suspended license says it is impossible to drive a car because driving a car without those things is illegal. The American who has no insurance or driver's license will say it's possible to drive a car because "the engine works, doesn't it?"
All of this discounts that sometimes people make honest mistakes (I got busted for driving on a suspended license, but the reason my license was suspended was because of an unpaid ticket from a few months prior. But the cop could see that it was marked as paid. What had happened was somewhere in the bureaucracy, the people who got the money for the ticket failed to inform the DMV that I paid the ticket (and I paid it close to the deadline) so they never removed me from the list of people who should not have their licenses suspended for unpaid tickets... So I had no idea (stuff like this happens way more with insurance as it's easier to miss a payment and notification of a lack of payment).