You should be able to start both streams from the same device, as long as you use different accounts to start the game. No one account can play 2 games at once. Instead, you can set up family sharing across your Google Family. For more information on this, check out this article:
https://support.google.com/stadia/answer/9575790?hl=en&ref_topic=9495322&dark=1Alternatively, you can start the game from the Chromecast itself using the controller. Make sure the linking code is toggled on in the Google Home app, turn on your controller, and enter the code on the screen. You may have to sign in. For more information on that, check out this support article:
https://support.google.com/stadia/answer/9580352?hl=en&ref_topic=9495322&dark=1
-- Baldy
I am no Google official. My messages and opinions are my own, and should not be used to judge Google or their products. Silver and gold Guides are independent volunteers that like to help out on the forums.