The Stadia’s “wireless” controller will only be wireless when played on a TV via a Chromecast at launch. For everything else, you’ll have to plug it in using a USB-C cable.
One of the selling points of Google’s controller when it was unveiled in March was its seamless connectivity. Its own Wi-Fi connection stays constantly connected to Stadia’s data centres, meaning you don’t have to re-sync the controller to, say, your phone (if you needed to start playing the game on something portable after being kicked off the TV. Again.)
But it turns out that this system no longer the case, at least for the near future. A new explainer video for Google about how Stadia works contains contains a small-print disclaimer saying, “at launch, wireless play is only available on TV using a Chromecast Ultra.”
Source: IGN.com Google Stadia's Controller Won't Be Fully Wireless At Launch