What school of thought in anthropology can apply to "The Gift" theory by Marcel Mauss?
That would be structuralism. Marcel Mauss was a French sociologist who was the nephew of Emile Durkheim, who taught and heavily influenced Claude Levi-Strauss, who was widely considered to be the "Father of Structuralism." Structuralism is a school of anthropological thought that holds all social interactions in all civilizations are constructed and not natural. It also holds that the language that is used by that civilization has meaning because of the words that are used. The words represent symbols, and the symbols have meaning.
In Mauss The Gift, easily his most influential work, gifts are symbols of possession, not just material possessions, but chosen by the giver, and as such, given as a part of the giver to the receiver. This "prestation" sets up an obligation for the receiver to reciprocate and give a gift back to the giver that is equal in value. These gifts obligate the recipient to respond and are thus bestowed with a "magic."
Levi-Strauss took this paradigm of gift-giving, and translated it into three parts: 1.) people will follow rules, once they are established, 2.) gift-giving is the simplest way to establish social relationships, and 3.) the gift binds both the giver and the recipient together in a type of social relationship.
Marcel Mauss is one of the most important writers of modern French anthropology, and in his book The Gift: Forms and Functions of Exchange in Archaic Societies, he outlines the social and economic importance of gift exchange in pre-Capitalist societies. Central to his thesis are the ideas that the exchange of gifts creates a bond between giver and receiver, the receiver incurs a debt or obligation to the giver (including an obligation to return a gift), and the competitive nature of some gifts, such as the conspicuous consumption of the potlatch, which serves to display the wealth and power of the person hosting it. Gifts thus have a sort of power, which Mauss illustrates with the Polynesian term mana, and their exchange is a central part of what creates and maintains the networks of relationships in and between societies. The exchange of gifts is therefore not just a substitute for money in societies that do not engage in trade/buying/selling. Instead, the exchange of gifts is what French anthropologist Claude Levi-Strauss calls symbolic exchange, and gift economies are those in which the values of objects are negotiable and in flux--their symbolic values are often distinct from their use values or exchange values, and their value and meaning are constantly shaped via their exchange. (For instance, the costume jewelry your mother keeps in her jewelry box is worth something to her not because of its monetary value but because of all the associations it has with her relationship with her mother, who gave it to her. Similarly, when a coworker always gives inappropriate or too-costly gifts at the office Christmas exchange, it can be read as him flaunting his money or as trying to assert financial superiority over or create a sense of obligation in his peers.)
Mauss's work on gift exchange theory forms an important foundation for Levi-Strauss's structuralism, and it has influenced Marxist and post-structuralist scholars as well as the field of economic anthropology.
For more information, see:
Mauss, Marcel. The Gift: The Form and Reason for Exchange in Archaic Societies, 1990.
Levi-Strauss, Claude. Introduction to the Work of Marcel Mauss. 1987.
Baudrillard, Jean. Symbolic Exchange and Death. 2004.