Voter suppression in the United States has had a long, sordid history. Blacks were given the right to vote after the civil war, prompting southern states to implement literacy tests, poll taxes, basically any legitimate-seeming test to de-facto turn away black people at the polls. This got so bad that we literally wrote an Amendment (24th) to state "yeah you can't do this either".
What's fucked up is that this sort of thing STILL happens. This past Tuesday, a number of voters in Virginia received calls telling them to go to a different polling station from the one they were assigned. See https://theintercept.com/2017/11/07/virginia-voters-get-myst...
Stuff You Should Know did a really good episode on voter suppression in the US that I'd recommend.
Why do they want a NUMBER? Because names are too... human. They're messy, and they can change, and they don't fit standard forms. It's an impulse to purge the humanity from social identity that motivates establishment of national ID numbers.
Then we got income taxes and federally guaranteed pensions (Social Security) and medical assistance (Medicare/Medicaid) and a few other, smaller safety net programs. Furthermore, while there's plenty of fighting around the margins, the overwhelming majority of Americans are broadly in favor of keeping at least some of the Great Society programs, and that means we have to have a national registry of some sort.
It turns out that when the population is broadly distrustful of the idea of a national registry but broadly in support of a tax regime and social programs that require a national registry to function the stable equilibrium is a really really shitty national registry that everyone hates but nobody feels they can improve without getting shouted out of office. American politics is dumb.
The only thing everyone has is a person number, and the difference is that it is explicitly considered public information and you can get anyone's person number by just asking the Tax Agency. It's just a convenient way to keep track of people, not a way to actually identify yourself.
Not all countries have or want that.