Yeah, that is the control and switch assembly. The center part (4-position fan switch) was available by itself for several years on eBay. I cannot seem to find one at the moment. However, I did find several offerings for that same assembly in the link you posted at a much cheaper price, made in China, no doubt. But, I suspect the one you linked is also made in China.
Also, when you are looking at your battery from the front of the car, notice on top under the center lid there should be three 30 amp fuses..... likely green since green represents 30 amp. The one on the left when facing the battery, is for the circuit that goes to the Fan Control Module. That circuit provides for the out front fan operation thru the FCM. That fuse is prone to corrode due to over heating ... if it has been over heating it will be quite evident. The other two fuses are for the ABS. (this info has nothing to do with your issue inside the vehicle)
And, there are two speeds on the Fans out front. As you saw, there are three wires, brown is ground and the other two are for low and high speeds. The slow speed is the one that goes bad, either by wearing down of the brushes or road debris getting inside the brush guides causing them to bind. Oh, and sometimes the resistor for low speed burns out. I have never seen a fan not work on high speed.
And, follow the two wires from the AC Compressor over toward the battery. There under or near the starter is a "bundle" with a connector that contains two wires that go to the Alternator. That connection seems to easily become contaminated and cause loss of connection....... generally an oily messy place to work.
Lastly, some where out there is an Outside Temp Sensor. It is generally located under the cowl on the left side of the car back close to the bottom of the windshield. If that sensor is bad or the wires have been chewed by a mouse, the AC Compressor will not kick-in. Assuming the refrigerant level is sufficient, the fans out front will come on when the Ignition is ON, AC Button Pushed in and Fan Switch ON any speed. The engine does not have to be running. If I remember correctly, if the fans do not come on (assuming good switch, good FCM, sufficient refrigerant, etc.), then the outside Temp Sensor may be the problem...
Hope this helps!