Product Owner is not the same as Project Manager. Product Owner is similar to Product or Program Manager in software, though the Product Owner role is usually a subset focused on software development and does not include go to market and P&L.
SCRUM Master role is the closest to a Project Manager.
The SCRUM Master isn't on all agile projects, but the SCRUM Master can be any combination of rotating team member responsible for following up on team tasks and facilitating SCRUM rituals to a full agile coach responsible for team coaching, organizational-wide systems/process analyst and change agent. Sometimes they get saddled with traditional project management deliverables as well.
The product owner, however, controls the backlog, and the yes/no in what defines the product is most often the most important (and difficult) position of the PM.
Right, Product Owners (like Product Managers) define the "what" of the product, effectively guiding the team on what market problems to solve and for which customers.
That's different from the Project Manager who works on resourcing and timing of a project.
SCRUM Master role is the closest to a Project Manager.