The problem of scaling is defining one Product Owner per Scrum Team, this behavior tends to lead to confusion.

As the product scales, more Scrum Teams can be added. But it will still remain one Product Owner. In theory, the Product Owner can work simultaneously with nine Scrum Teams, personally I haven't seem that yet.

From my experience, Product Owners can work efficiently up to five Scrum Teams. For more than that, I believe a scaling framework should be added. My preference is LeSS because it's light and simple. But there are many others.