The customer-expectation of a brand-product is based on the perception, the history and the presence of that brand, a well-crafted association and experience that is funded with the profit of the product.
The private-label product is cutting this funding. It will not finance the supplier-brand and it sets its own expectation and quality-thresholds on the product. The supplier accepts that and waives his control to get the volume-order and fill production-gaps in its factory.
In this kind of deal there is no direct relationship between the original brand and the private-label product anymore.
Edit, for clarification: It's not said that the potato-chips will be exactly the same. The minimum requirements of the brands may not match with the minimum-requirements of the private-label, i.e. for the private-label a different grade of potato, oil, salt might be used.