Why do we call quantum physics an ab initio model?
Quantum physics is called an ab initio model because there is not a more fundamental description of the natural phenomena it attempts to explain.
Why do we call classical mechanics an ab initio model?
This satement is odd to me. I do not consider classical (continuum) mechanics to be ab initio: rather, the equations of classical mechanics comprise constitutive relations that tend to relate averaged values of fields and fluxes. Constitutive relations, by definition, are not fundamental.
Is classical electromagnetism an an ab initio model (and if it is one, what are its first principles)?
Assuming we include relativistic electrodynamics in this discussion, then yes I would consider classical electromagnetism to be ab initio. It accurately describes how a fundamental physical force (electric charge) behaves. There are no more fundamental physical quantities to breakdown. There are no requirements for special frames of reference or to average field or flux quantities.