It’s beyond suspicion now. In 2015, Carly Simon said the second verse is about him. But to your point of his vanity:

Simon tells People, “I have confirmed that the second verse is Warren,” adding, “Warren thinks the whole thing is about him!”

Wikipedia claims, according to the podcast episode here, the 2023 appearance was unnecessary because Dick Tracy becomes public domain in 2027. No idea how true that is, but if it is true, that adds an interesting wrinkle to the story.