It really depends on the nature of the story. In a more fictional or children - friendly 'comparing cocks story', there could be other talking animals as characters. They could be friends or foes of the cocks. For instance, a wise old owl might be a character who gives advice to the cocks about the futility of excessive comparison. Or a mischievous fox could be trying to take advantage of the cocks' preoccupation with comparing themselves.
I'm not sure specifically which 'comparing cocks story' you are referring to. It could be a story about comparing roosters in a farmyard context, perhaps about their size, strength, or behavior.
The phrase 'comparing cocks stories' is rather ambiguous. It could be a play on words. Maybe it's related to comparing tall tales or exaggerated stories in a rather strange or humorous way. Without more context, it's difficult to determine its exact meaning.
Stories about friends comparing their knowledge in different fields are also great. For instance, in history, who knows more about a specific era, or in science, who has a deeper understanding of a particular theory. This kind of comparison promotes learning and friendly debates.
First, be aware of the inappropriateness of such topics. When in a conversation, if someone brings it up, firmly change the topic. For example, start talking about a new movie or a book you've read.