Chromecast

That's just Netflix "Casting" to a supported device. i.e your Chromecast. The Google Home App has very little to do there.

The Apple TV app does not support casting to a Google Chromecast device. However, if your specific Google Chromecast device supports Apps, you may be able to download the Apple TV app directly to the Google Chromecast device.

Same Link as above-> https://support.apple.com/guide/tvplus/google-tv-apd093a1a300/web

If it doesn't, the your only option is to use the Chrome web browser from your phone, log in to the Apple TV website, and cast it that way.

https://streamingclarity.com/cast-apple-tv-to-chromecast-options/

What hardware device is connected to your TV exactly?

What device are you "Casting" Netflix from exactly?

The Google Home App controls devices like lights and speakers and thermostats, it doesn't really have anything to do with Netflix.

Netflix can Cast on its own to a Chromecast capable receiver, so not sure how the Google Home App fits here.

If you have a Chromecast with Google Tv device you can download the Apple TV app to it. See here: link-> Watch on Google TV - Apple Support

The app or hardware device?