I currently have a Sony Android TV from 2015. A couple of months ago it started flickering badly and the repair was assessed to be prohibitively expensive, so I got a new one, another Sony Android TV, a 2020 model this time.
Is there a way for all my apps and data to be transferred over from the old TV to the new TV? Android phones have a new transfer feature on setup - is there something similar for TVs?
Not on setup. Use the same Google ID when you setup and the TV should be "essentially" the same. If you've sideloaded any apps, you'll have to do that again, but any "missing" Google Play Store apps can be reinstalled manually. Unlike a phone, there isn't a lot of "local data" on a TV to worry about.