Is there an etymological relationship between the words "Cohen" (Jewish priest) and "Khan" (Turkish or Mongolian ruler)?

Maybe way, way back (as in the time of the development of the Hebrew and Indo-European languages). But in the Semitic alphabets, the spellings of the two words are based on completely different roots (Cohen is K H N, Khan is H A N).

Khan is a popular surname among Pathan Muslims, but has nothing in particular to do with the Mongols.
As for the German 'Kahn', it is of Jewish origin and means 'priest' in Hebrew. It is also often spelled as Kohn or Cohen.

Khan is not a Mongol surname. It is a title used by the ruler, like 'king' in England. The difference is that in England and elsewhere, there are many people with the surname 'king', even though they are not descended from kings.

https://en.wikipedia.org/wiki/Kahn

https://en.wikipedia.org/wiki/Khan_(surname)