This is very interesting! First I would like to thank Fuubuu to bring up this subject. Even though I often read and hear about this usage, I have never thought about how to use this.
This might become too technical but I want to share something more of what I thought. I think Toritoribe-san's answer is spot-on. It (this usage) must be 同格. And this is what I summarized (at least in my head).
When using this particular 同格 form, "A の B", "A" is people's name or people's pronouns just like Toritoribe-san said. "B", I think, has to be な-adjective (形容動詞). All example we mentioned above are nouns but also な-adjectives. For example, we can say "ねずみはバカです。[(sorry, mouse!)]"(バカ is a noun here) but we can also say "バカな人" (バカ is な-adjective here). Likewise, we can say "エッチな人", "意気地なしな人", "わからず屋な人", "ケチな人" which indicate all these words are な-adjectives. We don't say "お兄ちゃんの学生!" (学生 is a noun), but we say "お兄ちゃんのいじわる!" (いじわる is a noun but also な-adjective). Of course, not all な-adjective can be used in this form; B has some sort of negative connotation in it just like Toritoribe-san said.