One of the most popular metaphors for describing what precisely the point of having a Product Manager is that they are the “CEO of a product”. Some people really click with that description, myself among them, and others reject it utterly.
I suspect that the divide comes not from the wide variety of ways that the role of Product Manager can be implemented in different organizations, but rather the differing perception of the role of the CEO. If you conception of a CEO is that of an authoritarian, the person who tells everyone what to do, The Boss, then clearly a PM is not the CEO of a broom closet, never mind a product. A Product Manager has no direct authority over any of the people they work with, and thus only indirect influence yet full responsibility over the final product, therefore, they cannot be the “CEO of a product”.
If instead you perceive the CEO as the person in the middle of the panopticon, the one who gets to see all the disparate parts of the organization and consider the CEO’s job to be leadership, unification of the warring factions, and vision, then we come much closer to the role of the Product Manager.
I know which reality I want to live in.