How are the beauty and makeup industry examples of capitalism? Please relate to Marx's theories and Adorno and Horkheimer's theories.

If we use the Oxford English Dictionary's definition of capitalism: "An economic and political system in which a country's trade and industry are controlled by private owners for profit, rather than by the state," then the answer to the first part of your question is very basic. In the global market, the vast majority of beauty and makeup corporations are privately owned and run for profit, thus making them examples of capitalism.
To expand into the philosophies you cite: Marx wrote in Das Kapital, “Here individuals are dealt with only as fact as they are personifications of economic categories, embodiments of particular class-relations and class interests.” Both the beauty and makeup industries are excellent examples of this, as they are extremely niche-oriented, targeting low-income, middle-income and high-income individuals very differently, as well as various ethnic groups, and having an interest in both showing and maintaining the class differences via the physical representation of the classes' members.
Adorno's concepts of "false needs" and "commodity fetishism" obviously directly apply to these industries as well. Does anyone actually need 50 lipsticks or a $450 moisturizer? But the supposed status conferred on someone who can afford them is actually perceived as worth more than the items themselves. And, the planned obsolescence of the items promoted by the industries keeps the cycle repeating.
Adorno's collaborator Max Horkheimer's work on "materialism" also can be applied directly to industries that equate happiness with acquiring more and more possessions. The beauty and makeup industries are expert at exploiting the human need to appear attractive to other humans by offering material "solutions" to problems people sometimes are not even aware they have. "Just one more wrinkle cream, one more exfoliator, and you'll be happy with your appearance and, by extension, your life" is their very effective pitch.