As far as I can tell, birth certificates are issued by states and name changes are done through states, so it isn’t clear to me how a name is a federally issued ID or even an ID.

While I am sure the bureaucracy surrounding such discrepancies is enormous, I’m not for sure I understand how it relates to a discussion about national IDs. A passport is effectively an optional national ID, so none of these issues would be solved by a national ID.

It's worse than that. Birth certificates are issued at the country level.

I think you have a typo and mean county?

And I'm guessing it varies by state how local the issuing authority is given that counties are barely a thing in some locales.

Except where they aren’t.

Huh? A name is something which is on a federal ID, obviously it is not in and of itself a federal ID.

The Social Security Administration (SSA) issues IDs at the federal level, which are applied for using state or county issued birth certificates (as well as other ways). They sometimes make clerical errors. Those errors can propagate into other federal identity databases, such as the taxpayer record system operated by the Internal Revenue Service or the passport database operated by the Department of State.

The linkages between these systems are fragile or nonexistent, and if you are not careful about manually propagating changes between them you can end up in an inconsistent state which requires court action to rectify.

Passports are often used in lieu of social security cards because people are (correctly!) leery about using their SSN directly. I know people who have no intention of traveling internationally who have gotten passports because it’s extremely convenient to have a federally issued photo id.