From the point of view of the Scrum team, the Scrum product owner calls all the shots about the product—most importantly, what features add more value than others, whether something is sufficiently complete to mark it “done” and so on.
However, that’s not really what it’s like from the product owner’s perspective. Scrum is centred on the development team and the language (“product owner”) is constructed accordingly. Running a training course for product owners recently one of them said he “felt like a servant with ten masters”. It’s a good analogy. While the product owner may theoretically have the authority to make product decisions, the reality is that they need to balance many conflicting and contradictory demands.
Stepping back from the narrow Scrum dynamic, the product owner doesn’t own the product at all. They are its caretaker, or foster parent, perhaps. The product may well live on beyond the time of that particular product owner’s tenure. It exists not for the benefit of the product owner, but for the users and other stakeholders. They are present for this period, but it must will grow and evolve beyond that. Product ownership is therefore a particularly demanding role.