Dose any branch of the field of economics have a term for when producers cease to compete at the production level, but continue to compete fiercely in marketing. The example I have in mind for this is the sports shoe industry (Rebok, Nike, Adidas, etc.), where production has been subcontracted to East Asian companies that specialize in finding the lowest cost producers in China, Vietnam, Indonesia, etc., where the same manufacturer can be in the business of making shoes in the same factory for several companies (Rebok, Nike) at the same. And what precisely are the theoretic conditions that enable this to happen? Are production costs that their absolute lowest so that no production competition is possible?
The term you are looking for is product differentiation. This is the name we give to a class of models (most prominently, the Hotelling linear city and Salop circular city models) where firms may have identical production costs and competition is about pricing and product positioning. In these models, we assume consumers preferences vary - some people prefer Nike to Adidas a lot, some by a little, some are indifferent and so on. Consumers will pay more for a product that is "closer" to their most preferred type, and firms decide where in the spectrum of preferences to position their product. This positioning can take the form of how you design the product, but also via marketing and advertising where firms may try to create different kinds of association with their product.
In the product positioning decision, firms often face a trade-off between positioning a product where it can closely match the preferences of the most consumers but where they are likely to face more intense competition from rivals, or to position farther from some consumers but also farther from rivals. Frequently, the best strategy for a firm will be to differentiate themselves from their rivals and develop a product that is really attractive to some consumers but not very attractive to others.
For this kind of competition to take place, you need a product that can vary on dimensions that customers care about. It's not strictly necessary that production costs are at their minimum, since firms can have different costs in these models. But the effects of product differentiation are probably more salient when firms are not also competing on cost. The models I described above are technically forms of horizontal product differentiation, where consumers will choose different products even when they cost the same. There are also models of vertical product differentiation, where consumers agree some products are better than others, but vary in how much they are willing to pay for the "better" product. Horizontal product differentiation is a good description of Nike versus Adidas, while vertical product differentiation is a good description of something like BMW versus Kia.