Strong product owners controlling the workflow and prioritization of the team certainly feel like management when you're working under one.

What makes it a hard call is that, in an organization that's trying to be engineering-focused, the product owner typically has as much say in what work you'll be doing as the person the org chart says you report to. (They have the most relevant expertise - a "people manager" has to spend a lot of their time managing people, while the product owner's sole and full-time responsibility is understanding what work the product needs.)

They shouldn't. Otherwise, there's nobody to pushback against stupid client requests. Product owners should be managing the product, not the employees. Having an employee manager to butt heads leads to a better overall product because it leads to better discussion between the devs who know the system and the product owner who is helping to flesh out requirements.