Yes, you CAN connect Kinect to an original Xbox 360 (Fat), but you need the Kinect AC Adapter / Power Supply Kit (sold separately).
Kinect Sensor │ ├─ Proprietary Plug → Kinect AC Adapter (splits into USB + Power) │ │ │ ├─ USB → Xbox 360 Rear USB Port │ └─ Power Brick → Wall Outlet
Direct plug into Orange AUX Port (dedicated power + data). No adapter needed.
Without the adapter, the Fat 360's USB ports cannot supply the 12V/1.5A the Kinect motor needs.