I think it sets the wrong expectations for those product managers but in terms of their duties and in terms of collaboration with their team members.

I've seen product managers who thought they were _managers_ actually in charge of the people on the team and this led to frustration. OR they thought they were decisions makers who had some measure of authority instead of just another member of a team with a specific role. In general collaboration with

And when I think about the product managers, I've worked with who described themselves as "mini-CEOs" they lacked basic product management skills. For example: focusing on the what over the why, being overly process driven, and lacking a cohesive vision for what the product will be long term.

I can't tell you correlation-causation here. Whether the description of "mini-CEOs" attracts people who are bad at product management or whether telling someone their job is to be a "mini-CEO" causes them to behave poorly.