I don't think you can. On the other hand, if you register a Google account, use a secondary user on your phone to login, install the app and activate the Chromecast, I think you can subsequently use it without the Google account. Delete the secondary user once you're done with the setup. You wouldn't have given Google any useful data and you'd have cost them some.
My Google home and Chromecast at the moment are necessary evils because when I sit down to watch something I don't want to have to worry about whether it'll work with x or y, whether I'm getting good bitrate etc
Would however be really cool if I was able to cast my screen, cast YouTube etc from my Linux laptop
Have been able to do it in the past using chrome but I already need to keep two browsers installed don't want a third
Just a thought, but since someone else in the thread said you can stream to Chromecast via VLC, you can desktop capture natively in VLC and stream that to your Chromecast. I can't remember if the native capture can do sound or not, but if not, you can instead use OBS virtual cam (you'll need v4l2loopback for the virtual cam to show up), and open that as a capture device in VLC. You should be able to attach an audio source to that as well. While I haven't personally tested it with audio, I have used OBS virtual cam with VLC before, and it worked flawlessly for me. If you can't find a more elegant solution, then it's worth a shot to try and see if it works
If it's actually possible when my Chromecast stops being usable I'm putting Linux on it and using it to run some light weight projects (someone mentioned you can Linux them)