The difference is that every country with any sense actually keeps track of its citizens. Consistently. Because when you say it out loud, it sounds ridiculous to be a government with no idea who your people actually are.
For half a century social security cards had "not for identification" written on them because they are really bad for ID. Didn't stop everyone from using them as ID because there was no other option. The Social Security administration was the only government agency actually tasked with tracking citizens for their entire lives so they did that. Poorly.
If you want to understand why requiring any state ID is voter disenfranchisement in the US just consider what it takes to get a state ID as a fresh adult. Two recognized proofs of ID. Basically birth certificate, student ID, social security card, or other state ID. If you don't get a student ID, didn't go to high school, or your ID from there simply expired it can't count. So its basically requiring birth certificate and social security card.
Now thats where you might not recognize how weird the US is. Nobody is required to have those. Hospitals do the paperwork to get them issued when you are born there but for every child not born under the supervision of a doctor (way too many to this day in the US) your parents do need to actually apply for those. Of course that only matters if you are filing taxes to begin with. Which is disproportionately whites. A ridiculous number of people exist outside the tax system because... the government doesn't know who its citizens are. It doesn't keep track of anyone in a central way. If you IRS never started getting pay stubs from your employers they might not even know you exist.
This is in addition to having the means to get to a DMV, pay the fee to get an ID card (I think it was $25 when I got one to fly to Texas when I was 17?) and be able to go there during some obscenely limited hours of operation (some of the worst offenders I've seen are only open 10-2 on three weekdays). Then you also need a permanent address to get it shipped to.
For way too many people they are missing something in that chain, and if you find yourself without two qualified forms of ID getting a birth certificate or social security card where you never had one before can take months at least.
What might sound really bizarre to you is that there is nothing to really "catch" kids that fell through the system. School enrollment is completely decoupled from all the previously mentioned systems of tracking people - you don't need any major documents to enroll your kid in a public school, often not even proof of legitimate residency in the country. Schools want those but the laws are state specific and often simply "try to figure out who they are or just drop them in class somewhere I guess". So you can easily graduate high school with no government institution knowing you exist whatsoever, go to get a job, and find out your employer won't hire you without this backlog of paperwork nobody made sure you had.
This whole mess, as others say, disproportionately affects minorities and the poor. They are less likely to have parents who can afford hospitalization for the birth so they don't get the paperwork done for them. They don't pay taxes or work under the table and thus don't itemize their children or apply to get their documents to get tax rebates. They get them enrolled in school without having any documentation of their own. They go their entire childhood using health clinics and whatever their mom thinks is best to get through because they don't have health insurance and never show up in that system. Nobody in their family can afford a car so getting into the drivers registry isn't even on the radar for any of them.
There are millions of legitimate born US citizens to American parents without a paper trail because they live in poverty. Its the 1 in 4 food insecure kids growing up today. Their towns and cities are practically blighted with mistreatment by every institution that has any influence on their success. I live in central PA and see this all the time - neighborhoods can be 10 miles apart where on one side they don't have regularly working electricity and water while on the other end you have mcmansions and three newly built shopping centers composed of Whole Foods and Old Navy. One of those groups is going to have their learners permit at 16 and be preregistered to vote without even knowing it, the other would have to spend years of time fighting to form the paper trail to register to vote.
And its a legitimate fight. My parents lost my social security card at some point in my early childhood and it took 2 years and 3 visits to the DMV to get a replacement. Good thing that happened when I was 12 and not 18 or I would have missed upwards of two elections like that. And I was on the lucky side of the ID disparity - born in a hospital, records of doctors all the way through, proper paper trail in the public school system, and it still takes several hours of your life to get through the beuracracy to have a drivers license or state ID by the time you are eligible to vote. Way too many people don't have the paperwork or the time to try to use it as a barrier to voting.