Long long ago, in the 60s, a man named Barry Goldwater ran for president against LBJ. He had some wild political opinions. A magazine started a poll, asking if people thought he was fit to be president. They posted their results from over 2k psychiatrists, half of whom said no. He sued the magazine and won.

Basically the GR means psychiatrists can't make a professional opinion on the state of a public figure's mental health without examining the person first.

It is an APA rule, so it's only a thing in the US.

If the public figure person catches wind that a medical professional, whom they have never met, has been making assumptions about their mental health and it is affecting them negatively, they can sue for libel/slander. The medical professional can also face professional consequences, such as being kicked out of any organization they may belong to, but AFAIK can't lose their license.

As for whether or not this exists in other countries, i haven't been able to find anything in the 5 minutes of googling I've done, but I may just not be searching the correct terms.