In the TF2 Wiki, it mentioned that "two Snipers wielding the Huntsman or Fortified Compound can create an infinite stun loop without killing each other, commonly known as a "Staring Contest"."
However, the Wiki page for the Arrow Stab taunt kill does not mention that this bug has been fixed and has been almost lost to memory, and was a beloved part of the community back in the day as shown by the number of videos about it such as this: https://www.youtube.com/watch?v=b7XDbzxpGJM
To perform the Staring Contest bug, two players each from opposing teams equipped with the Huntsman/Fortified Compound must be present. Both players then taunt at around the same time with the bow. After both players are stunned, the taunt key is immediately and repeatedly pressed quickly over and over again until one player eventually tires out from repeatedly pressing the taunt button and loses the Staring Contest.
If you try to do the Staring Contest bug in live TF2, both Snipers do get stunned by each other but the loop does not happen. You can test this with a puppet bot with zero ping and no matter how much you try to mash the taunt key, you are not able to do the Staring Contest bug. It used to be much harder to do at higher pings. You can see a demonstration of the modern behavior with this video: https://youtu.be/9UMqfykEsP4?si=SYwfRnvJOL2yYqdr&t=41
I believe this bug was inadvertently fixed sometime around 2013 (or 2017?) based on patch notes, video comments, and historical videos related to the taunt. I still have yet to confirm when exactly was the bug fixed. What I am sure of is an unrelated change fixed it. If anyone here knows when it got patched, please do tell since that would be useful information and it could be checked by running older builds of TF2.
If you ever wondered why no one seems to do this anymore, here's your answer. It was shadow patched out of nowhere in the mid-2010s when Valve was adding more taunts to the game.