-Why do we call quantum physics an ab initio model?-
Quantum physics is called an ab initio model because, as opposed to (semi-)empirical approaches, no input besides a key first principle is required here. Such a principle is comprised by the Schrödinger equation.
-Why do we call classical mechanics an ab initio model?-
Similarly, classical mechanics is fully built upon a principle equation(s), the Newton equations of motion, which are enough to describe all mechanical properties of a classical object. Interestingly, Schrödinger equation may be considered an equation of motion, too, if “derived” using a classical wave equation and the de Broglie hypothesis.
-Is classical electromagnetism an an ab initio model (and if it is one, what are its first principles)?-
To my mind, classical electromagnetism is an ab initio model as well, since it fully relies on the Maxwell’s equations, same as quantum and classical mechanics rely on respectively Schrödinger and Newtonian equations. Maxwell’s equations would then be considered the first principle here.