I love this question. Supposedly, Walmart does a significant amount of profit from their house brand which is, inspired by their suppliers in a similar way to Amazon.
As a Costco shopper, I am also quite likely to grab the Kirkland brand instead of the competitor.
Costco is maybe not the best example, because nearly every Kirkland product is a main name brand with a different name. It's almost never the case that Kirkland coffee (often made by Starbucks) is on the shelf next to an identical/equivalent coffee from, say, Folgers. Costco often only stocks one type of each product: they have Kirkland batteries (Duracell) but no Energizer (though they do sell other Duracell batteries at a higher cost).
Most supermarkets, on the other hand, work with manufacturers that aren't name brands, and almost exclusively sell their store brand alongside name brand products.
While I don't know what makes Amazon's practices different from what supermarkets do, my assumption is that placing two products adjacent on a shelf is very different then highlighting and promoting the store brand above and ahead of the competition. The store brand doesn't appear exclusively in recommendations or ads or come with a "Preferred" badge. The store also doesn't use sales data to decide which products to make a store brand for, with the goal of overtaking the name brand.
From what I understand most of those "house brand" products are made by the same factories of the on-brand stuff, probably but not necessarily with the same recipe and QC standards.
Product branding has almost no relationship to which company actually made the product, consumers are intentionally blinded to this information. I would appreciate regulation requiring products to be branded with the factory that made it; I think this would be a huge win for both consumers and workers but it would be firmly counter to the interests of major brands with a lot of political influence. Also many consumers would probably oppose this against their own rational interests because they have personal attachments to brands.
Grocery stores used to be marketplaces. Then in-house brands came along. The current iteration of grocery stores is Aldi, where everything they sell is an in-house brand.
