XInput and DirectInput
Also called XInput · DirectInput
XInput and DirectInput are two Windows interfaces games use to read controllers. Many modern PC games only support XInput, which is why a controller that works everywhere else can be ignored by a game.
A browser reads controllers through its own interface, so passing the controller test proves the hardware works but not that a particular game will accept it.
In practice
XInput is the reason an Xbox controller just works in most PC games. Controllers from other makers often support it by pretending to be an Xbox pad. DirectInput is the older interface, still used by some controllers and flight sticks. Games that only read XInput ignore DirectInput devices completely.
How to check
If a controller works in the controller test but not in a game, check whether it has a mode switch or button combination for XInput. Many third party controllers do. Steam Input can also translate a DirectInput controller for games launched through Steam.