It's not recognizing your Xbox 360 controller because the Xbox One adapter does not work with Xbox 360 controllers. As Chary says, 360 controllers use a proprietary connection, it's not the same as an Xbox One controller's. You need to have a 360 wireless adapter if you want to connect your 360 peripherals to your PC, the Xbox One adapter can't do that.
EDIT: Alternatively, you could buy a wired 360 guitar. Those are your only real choices.