When is a food considered a delicacy?
I found your question very interesting as, even though we know there are delicacies, I never thought about how something came to be considered a delicacy.
The following excerpt from an article in  psmag.com  sheds a little light on the subject. The article in its entirety is very interesting and demonstrates how the same thing may or may not be a delicacy today and the opposite tomorrow.

Cultural influences lead to the difference in the habitual consumption of certain foods and in traditions of preparation, and in certain cases can lead to restrictions such as exclusion of meat and milk from the diet. Cultural influences are however amenable to change: when moving to a new country individuals often adopt particular food habits of the local culture.
Social influences on food intake refer to the impact that one or more persons have on the eating behavior of others, either direct (buying food) or indirect (learn from peer’s behavior), either conscious (transfer of beliefs) or subconscious. Even when eating alone, food choice is influenced by social factors because attitudes and habits develop through the interaction with others. However, quantifying the social influences on food intake is difficult because the influences that people have on the eating behavior of others are not limited to one type and people are not necessarily aware of the social influences that are exerted on their eating behavior.