Here is the solution
What you need to do is exit the TV app, open the Settings app, select Device -> About -> Software Version and then press the Check for System Update button. What you will probably find is there is a new update available; seemingly the Fire TV is very slow at notifying users about the need to update the operating system.
When the update is found, the Fire TV Stick will start downloading the newest OS firmware automatically. Once the download is complete, press “Install System Update” to update.
After updating to the latest OS version, you can then go back to the Apple TV app and sign in again. The login process should now complete successfully.
Try signing from the settings option, not from the library option