Has nothing to do with whether he’s American. An Australian living in, I don’t know, the UK or Sweden, but on the payroll of the government of Russia, has to declare that fact to the government of the country in which he resides. Otherwise he can be considered a spy. And no, “being on the payroll” is not limited to being an official employee of or even receiving payment from Russia. Depends on the laws of the host country but in most places it implies simply “working at the direction of” or “receiving compensation from”, including non-monetary compensation. Like, say, hacked emails that you can use to promote your organization.
Look, I am a US civil servant. When I travel internationally I have to use a special passport. It’s maroon instead of the usual blue, and it has a stamp in it that states that I am a representative of the US government and am on official business.
And this is exactly why. When I present it at a border crossing, that is an official legal declaration to the country I am visiting that I am a declared agent of a foreign government. If I attemp merely to enter the country using my personal passport I am committing an arrestable offence.