With authority comes responsibility for your actions. Without responsibility, no authority. The product manager is a manager in name only, and product owner even less so.

That doesn't mean you can't have several direct reports. The classic matrix organization for example. But it means semi-managers without real responsibility have no real mandate for doing a good job at the slightest hint of trouble.

If there's a conflict of interest, it needs to be discussed based on merit, not based on who has the bigger authority.

If there's no agreement, it needs to be escalated to somebody who has the authority (manager). But IME this doesn't happen very often.

I like this model, because the default position is that none of the engineering, product, process is the "master", so you need to negotiate. If one of the roles also has reporting authority, that automatically skews the decision making towards yielding to them.

So what is his job then?

NIMS, the National Incident Management System, talks of ICs having between 3-7 direct reports, when there is a need to be connected to what they are doing, because beyond that, you can't reconcile things easily.