The question on the differences of the naming has caused a lot of confusion. I will try to simplify as much as possible here.
The product owner is a well defined role in scrum, the scrum guides defines that role as follows:
The Product Owner is accountable for maximizing the value of the product resulting from the work of the Scrum Team. How this is done may vary widely across organizations, Scrum Teams, and individuals.
The Product Owner is also accountable for effective Product Backlog management, which includes:
Developing and explicitly communicating the Product Goal;
Creating and clearly communicating Product Backlog items;
Ordering Product Backlog items; and,
Ensuring that the Product Backlog is transparent, visible and understood.
The Product Owner may do the above work or may delegate the responsibility to others. Regardless, the Product Owner remains accountable.
What about a product manager?
I like how Melissa Perri describes that the job role is called a product manager does the role of a product owner. She even brings the example from scrum alliance “the Product Owner is typically played by the customer, or customer representative, such as a product manager.”
And to achieve the responsibilities of a product owner well “A good Product Manager is taught how to prioritize work against clear outcome oriented goals, how to discover and validate real customer and business value, and what processes are needed to reduce the uncertainty that the product will succeed in the market.
Without this background in Product Management, someone can effectively go through the motions of Product Owner role in Scrum, but they can never be successful in making sure they are building the right thing.”
There is another layer I see to this as well. The product manager role in tech especially is still evolving and used differently across different organizations. So I will focus only on what I consider a product manager in a more Agile setup.
I would consider the threshold to be three teams. Beyond that threshold it is hard for a single product manager to function well. Then a lot of companies start getting team level product managers (associate product manager or a product owner) and a product manager at the higher level. And the point at scale to keep it simple, imagine writing epics or product objectives that are aligned amongst the product managers and then each team can break down into stories.
What if it is an even larger product? More than 10 teams? Then we can look at roles such as chief of product and that is alignment done at larger epics to start breaking down.
Word of caution to stay Agile. The team and the product managers as part of those teams need to stay close to customers and not separate the value discovery and creation cycles. It has to be implemented in practice and not just simply say that we are user or customer centric.
So in short, scrum needs a product owner in the scrum team, a product manager can fulfil this role, yet scrum does not define how he goes about doing his job.