I just recently watched Deep Space Nine again, and towards the end of the show Chancellor Gowron gets challenged by Worf to a fight to the death effectively for the Klingon Empire. That made me wonder, was there ever any culture that really had a "if you challenge the King, the King MUST fight you for the throne" rule or is that exclusively in the realm of "honor culture" fictionland?
I'm not 100% sure on this, but I believe this did exist in the classical and medieval period, specifically among the Germanic (Vikings are included here) tribes and later feudal societies. Roman writers remarked on the barbaric nature of Germanic law, such as determining the guilty via a duel (yes, trial by combat was a thing). Of course, once the Germans had absorbed Roman law and culture, this idea slowly became a thing of the past.
This would rarely be challenging someone like the leader of the tribe/the King/Emperor, but rather would be used to solve border disputes between families and other such affairs. Due to the Germanic origins of the HRE and France, I believe it was covered in their legal codes (prescribing the weapons that one could use, what situations was it suitable to use trial by combat, which was only when there were no witnesses for either side and etc) which lasted for quite a while, all the way up to the 15th century.
However, challenging the King/Emperor was not a thing, as one could never find the King/Emperor guilty; no doubt they would have legions of witnesses on their side, and you would be accused of fraud and deception; if you were politically powerful enough to challenge the King/Emperor for the throne, you would likely be using your armies to fight for it instead, or via assassination.