Julius Caesar was assassinated for wearing red boots. Barrack Obama was lambasted for wearing a tan suit. Purple is associated with royalty, white is associated with clergy, and gold is associated with superiority. Red and Blue have powerful meanings in America, and so does Green. Which colour is the most political in your opinion?
Drag is not going third person. Drag is using first person neopronouns. Drag is happy to teach you about grammar structures you're unfamiliar with. Drag prefers drag's pronouns over gender neutral pronouns like "I" because drag's gender is not neutral.
In Japan, there are multiple first person pronouns depending on the speaker's identity and indeed the specifics of their current social situation. Like watashi, boku, and ore. Drag is doing that.
Also in Japan, if someone refers to themself with their own name, that's seen as childish or arrogant.
If your first person pronoun is the same as someone referring to you in third person, then you are referring to yourself in third person every time you refer to yourself in first person. And since the idea of a first person neopronoun is exceedingly rare, most people will only think of you as speaking in the third person.
Drag is not referring to dragself by drag's name. Drag is referring dragself by the pronoun "drag", while drag's name is Dragon "Rider". Drag would like to remind you that drag's name is perfectly visible on your screen and you can read it whenever you like, so you don't need drag's help to know what drag's name is.
You refer to yourself as "Drag", which I understand is short for "Dragon". And given your proclivities towards dragons, I have to assume this is intentional. So, yes, you are referring to yourself by your name, or at least a nickname.