Thanks for the warning.
This does sound odd. Why would they need you to go to the play store and download something, in order to deal with your Amazon account? It does sound odd.
YouTube has many videos of this and other scams. The Amazon/iPhone scam is fairly common. Not Nigerian Prince common but maybe someday.
“Any Desk” allows them to access your accounts without your further permission. That includes your bank account.
Never ever ever install “Any Desk”.