I will bet there is no Kentucky law that orders county clerks to issue marriage licenses. They do so by custom in Kentucky and all over America. They also do it to bring in some money.
IOW it is traditional, logical and common sense for Kentucky county clerks to issue marriage licenses but there is no Kentucky (or Federal) law demanding they do so.
So I don’t see where Judge Bunning gets off demanding anything from clerk Kim Davis
Actually, there is a very specific law. The backup is a Justice of the Peace, IIRC. The county clerk is required to sign that the license is issued under their personal authority.
The rules for who may have a valid license specifically identify man and woman, and disallow homosexual “marriage”.