I have a 6 yr old Sony smart tv. I subscribed to Apple tv but am unable to watch shows

Sony’s oldest TV models that support the Apple TV app seem to be select models (not all models!) from 2018 [A9F series, Z9F series]. For supported models Sony TV, you’ll have to find and download the Apple TV app from the on-device Google Play Store; I doubt it is pre-installed there.

Unsupported models TV may connect an external HDMI streaming device that supports the Apple TV app. That can be an Apple TV box, Roku, Fire TV, Xbox, PlayStation, Nvidia Shield, ‘Chromecast with Google TV’ dongle, or one of select cable TV boxes that support apps.

Watch on these smart TVs and streaming devices - Apple Support

Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support

All the ways to watch in the Apple TV app - Apple Support

Apple TV app user guide for smart TVs and third party streaming devices - Apple Support