At last month's launch, the Stadia wireless controller only supported Chromecast Ultra that came with the bundle (although there seemed to be some exceptions). But this will change in the coming year. According to 9to5Google, the wireless controller for the game streaming service will support other devices like smartphones, tablets, desktops, and laptops. An updated Google Store listing hints at Stadia supporting wireless connectivity on more devices in 2020, possibly through a software-side update.
The current workaround is that Stadia supports third-party controllers like those for the Microsoft Xbox One, Nintendo Switch, or Sony PlayStation 4. Google has more plans for its Stadia platform, including expanded support and new features. All of which should also come in the new year.