Originally, in order to use the Stadia controller on your PC, it required the use of a USB Type-C connection. Now, after nearly five months, you can connect the Stadia controller completely wirelessly! However, it will only work with Stadia, and can not be used for Non-Stadia games.
In order for you to use the Stadia controller with Google Stadia, you will need to link the two via your Wi-Fi network, instead of directly connecting it to your computer. So this is why you are unable to use the controller with non-Stadia games.
Last year, Stadia announced that the controller would have wireless connectivity, but it was not available at launch, only allowing you to use the Stadia controller only if it was directly wired to your computer. But now, with this announcement, the feature is finally available.