Is a name a noun or an adjective?

Noun noun A word (other than a pronoun) used to identify any of a class of people, places, or things (common noun), or to name a particular one of these (proper noun). - ODO

The point of interest here is that a noun is a word, not the concept or the object. Consider it a label if you wish. If you change your name, you simply switch labels. You then have a previous name (a noun) and your current name (also a noun).

In your sentence "A noun is a person, place, thing, or idea", the word is should be understood as the following:

be verb 3.3 Represent. ‘let A be a square matrix of order n’ - ODO

That is, a noun represents the person, place, etc, as opposed to being those things ontologically.