We combine transport experiments, advanced ab initio calculations, and model analysis to determine the thermoelectric power in the two-dimensional electron gas formed at the paradigmatic oxide interface SrTiO3/LaAlO3. We demonstrate that contrary to popular expectation, quantum confinement does not enhance the thermoelectric power of the electron gas at this interface with respect to its corresponding three-dimensional case. Our analysis directly relates the thermopower behavior to band structure characteristics typical of the oxide heterostructure (i.e., on-site and intersite band splitting), furnishing general interpretive prescriptions to search for oxide heterostructures with improved thermoelectric capabilities.