If someone uses your name in all correspondence with you, and this is not necessary, then what part of the English language is the name?
What is your point Jay?
Stating the addressee in a sentence is called direct address. It should be punctuated like an adverbial phrase. Your example should be punctuated with a comma: "What is your point, Jay?"
The part of speech for people's names is:
a proper noun or proper name
They act grammatically mostly like other nouns (common nouns) but with some minor differences.