Update: June 23, 2021
The Stadia app has arrived for Android TV. As stated before, it is available for Chromecast on Google TV as well as Shield TV and several select Android TVs (mentioned in the original post). The app, however, showed up a bit early for the select TVs, but with a "Coming soon" screen.
Original Post (June 9, 2021)
Google Stadia cloud gaming service went official 18 months back. It allows users to stream games on their browsers, phones, tablets, and Chromecast Ultra. But the support wasn’t made available for Android TV. However, it is now finally coming for the platform so that users can stream games directly from their smart TVs running Android.
The support is currently arriving for select TVs, including the latest Chromecast with Google TV and NVIDIA Shield TV. Other than these TVs, the new Stadia for Android TV support is coming to Hisense Android Smart TVs (U7G, U8G, U9G), Walmart Onn FHD Streaming Stick and UHD Streaming Device, Philips 8215, 8505, OLED 935/805 Series Android TVs, and Xiaomi MiBOX3 and MIBOX4.
If you have any of the listed Android TVs, simply install the app and launch it from the “My apps” row on your TV, and press “Continue” at the opt-in screen. You can use a Stadia Controller or a compatible Bluetooth controller to play games.
As per Stadia’s official blog post, if your device is not listed above then you can opt into experimental support. “While this feature is still in development and not every Android TV OS device will work perfectly, you can now try out Stadia and play your favorite games on more screens than ever before.”
You can visit the full list of 180+ games available on the Stadia Store. Google has announced that Stadia on compatible TVs is coming on June 23.