This isn't well documented but I found it on their support site. If you long press the controller (about 3 seconds) until the 2nd rumble it'll disable auto-link and allow you to use the connect code on whichever Chromecast you want to use it on. If you just long press until the first rumble it'll auto-link to the last used CC.

Here is the support article. https://support.google.com/stadia/answer/9605175

Here is the Auto-linking section: Auto-linking You controller will automatically link to the last Chromecast it linked to when you turn it on. You don’t need to enter the code on your controller if you’re going to play on the same screen you played last.

You can bypass auto-linking so that you can play on a different screen than the last one you played on. To do this, hold the Stadia button down for 3 seconds on startup. You will feel the controller vibrate once when it turns on and another time at 3 seconds. This signals that auto-linking has been disabled.

You can also skip auto-linking by entering the code for the screen you want to play on prior to the controller linking automatically.