Each company will have desired traits in their managers, whether they go for the servant leader or the authoritarian delegator, their desired traits will limit who they view as suitable for managerial positions. There will be "manager types," and to people who have worked for that company for 10, 15, 20 years, that is the "manager type" for all companies, in their view.

There are also some traits that _are_ required for managerial work, certainly not enough to define a personality type, but without them, people could not function in a delegating, influencing, or persuading position. They're the traits that allow people to perform those functions.

I'd certainly be willing to concede the point that those aren't even personality traits, but skillful applications of personality that could be developed.

TLDR: I agree with you, but there's enough nuance in what "types" could mean that you're looking at a discussion about very subtle things that many people might not consider at first.