It's pretty much the exact same thing in the US. Even if you don't have to show a state ID/driver's license, you have to be on the list and usually need to show something. You can only get on the list by registering, which requires US citizenship.

The apparent source of this propaganda is based around the concept that some areas allow people with only a Permanent Residency but who have not yet gotten citizenship to vote in some local elections, like school board members, mayors, or on local referenda. This still notably doesn't apply to anyone who isn't a permanent resident or citizen nor does it allow permanent residents to vote for the president, or even moderately important figures like congresspeople or governors.