Apple - Stop unknown devices trying to connect to Apple TV
You can give AirPlay a password in the settings on your Apple TV. Only with this password you can connect.
- Open Settings
- AirPlay
- Change 'Security' to Password
- Turn off 'Require Device Verification'
In tvOS 13 there is a new option which should make this easier:
Settings > Airplay and Homekit > Allow Access
Change "Also Allow Nearby to Airplay" to Off