Tortured Poets' or Poets? Taylor Swift Meets the Apostrophe Police

I'm curious about opinions that this is part of a trope or pattern I sometimes notice in English: we can interpret "The Tortured Poets Department" as kind of a more euphonious way of writing "The Department of Tortured Poets."

While "The Tortured Poets' Department" is also valid English, it changes the meaning slightly from how we usually think of what a Department means. A Department doesn't belong to the group of people, instead it consists of them.

I'm not sure if I'm imagining that this is even a real pattern, or if it is, is there a word for it?