Were you able to verify with your father the exact part he used? I was looking it up, and the part appears to be made by Facet Spl in Italy, and VW sells the Facet part as VW Genuine OEM. The model # (as mentioned above) is 5G0698459 for the Facet and VW part. Looks like it's the same part from 2007-2018, so, nothing special about an 09.
Anywho, the above said, the switch appears fairly simple. I'm not certain how it works, but looks like it's a sensor that may work like an ABS sensor (detects position of a magnetic object) on the master cylinder. So, I did some searching and found somebody else on the MK6 forum (note brake light sensors on MKV, MK6, and MK7 are all the same) and this guy had a very similar problem -- almost laughably similar. He took it to the dealer, the dealer replaced the part, the part did not fix the issue, and the dealer left it at that - blamed his modifications for not being able to do a proper scan. His lights were stuck on all of the time, like yours, meaning the car could be started even though the brake light switch was not working.
So, without further adieu or drama -
HE was able to trace the problem back to a blown fuse. I don't know what fuse is linked to the brake light switch (ie - not the brake lights, as they work, the fuse for the switch may be blown). Have your dad check the fuses -- google it to see if you can track down the exact fuse location. I can tell you, though, I have a Haines Repair Manual at home, and it has a fuse diagram if you can't track it down.
Let me know if this turns out to be the issue. Also, read the post from 2012 for yourself:
https://www.golfmk6.com/forums/showthread.php?t=50189