That is largely not the case. I spent ten years making private label medical devices for CVS, Kroger, Target, and dozens of other stores.

There tend to be specialist manufacturers who fill the store brand niches. E.g. in pharma, close to 90% of the pills, tabs, and liquids sold in front of the pharmacy counter are made by one company, Perrigo, whose entire model is predicated on being a store brand supplier.

I don't think Kimberly Clark makes the store brand paper products, nor does P&G make the store brand beauty/cleaning supplies.