MacBook Air mid-2013 IOS 10.11.6 and compatibility with AirPlay

The only option shown under AirPlay Display is OFF because your MacBook Air has not sensed any other AirPlay capable device. Had it sensed an AirPlay capable device then it would have listed it right under the OFF option.

You have a SONY Bravia TV with AirPlay capability. Have your turned AirPlay to ON on your SONY TV? I know that you stated that they are on the same WiFi Network and that is also a requirement.

This forum is not for SONY Devices however I found the following from SONY's Website via a Google Search-

1. If AirPlay has not been configured, press the  (Input select) button on the TV remote control and select  (AirPlay), then select AirPlay & HomeKit settings and turn on AirPlay.
2. Make sure your Apple device is connected to the same network as the TV. 
3. If you use an iOS device such as iPhone or iPad: Open the Control Center and tap  (Screen Mirroring) on your iOS device.
4. If you use Mac: Click  (AirPlay Video) in the menu bar.
5. Select your TV. The display of your Apple device can be seen on your TV. 
6. If an AirPlay passcode appears on your TV screen, enter the passcode on your Apple device.

Hope that the above helps otherwise please refer to your SONY Manual or contact SONY support.

Axel F.