This is.. not quite true. Social security numbers are used this way (even if they aren't intended to be such a thing) and passports are essentially a national ID system. My passport is the only valid ID that I have - of course, I live outside the US and have to have the nearest US embassy renew it. (I don't qualify for Norwegian IDs yet, and my immigration card is a supplement to my passport instead of actual ID)
Not very helpful as a national ID if they are a relative rarity
Tax IDs are as close as we've got.
SSNs? Sure, they’re (optionally) used as an identifier. But they’re primarily used in the financial world, along with your credit score. Most state and federal agencies won’t require an SSN for official business (again excepting taxes & payments). A hotel will accept a Global Entry card for check in. UPS will accept a non-driver’s license to pick up a package. You can vote with a student ID in some states.
And don’t forget people who are undocumented immigrants or on F-1 and J-1 visas who don’t even have a SSN, but still need to open US bank accounts, pay taxes, get US driver’s licenses, and vote in municipal elections.
Identification is complicated in the US!
They're misused as an identifier. They've also been leaked to fuck and back.