The term ‘mayor’ by itself is not a proper noun. A proper noun is a specific name used for an individual person, place, or organization, and it is always capitalized. For example, when we say ‘Mayor Smith’ or ‘Mayor of New York City,’ ‘Mayor’ is being used as a title, and ‘Smith’ or ‘New York City’ are proper nouns referring to specific individuals or locations.
In contrast, when you use the word ‘mayor’ in a general sense, as in ‘The mayor addressed the crowd,’ it is not capitalized and does not refer to a specific name or title. In that context, it is a common noun. Proper nouns are specific, while common nouns can refer to general categories of things.