Some other users have also mentioned similar sign-in errors on this forum.
I haven’t seen a broad general resolution here though.
- Please verify that the Fire TV system software is up-to-date.
- Verify that the Apple TV app is up-to-date, if not done automatically.
- Verify that your router or firewall doesn’t block Apple authentication servers, in case you made custom changes.
- Verify that your devices and router use a good DNS, like the one from your ISP, which gets updated with all the server names that Apple might use. Again, only applies if you made custom changes.
- Verify that your Fire TV has the correct date, time, and time zone, as Apple may be more strict about that than other content providers (for TLS secure connection reasons).
- Try a restart of the Fire TV device.
Else I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/email/callback).