What is obviously false?

Every company I've ever worked for has had employee guidelines stating "don't comment to the press or mention the company on social media without approval"

Is this not a thing in the USA?

Its illegal in the usa to prevent someone from talking about their salary. Most companies don't have policies that expose them to legal issues.

I also think that its debatable if indicating your salary or admiting to the fact you work somewhere, counts as talking to the press or mentioning the company, the way most people interpret that phrase. I mean, i'll admit that its highly debatable, but for example putting your resume on linkedin usually does not count as mentioning the company on social media.

Companies have "legal documents" that employees sign that often are not legally binding. Say, for instance, if they openly contradict federal law.

And that is not a new thing in the USA at all.

I guess it depends on the exact wording but usually clauses like that refer to speaking on behalf of the company. Otherwise it’d be a ban on LinkedIn altogether for example.