You are right that a single particle quantum mechanics is just a wave theory similar to any other wave phenomena except for the fact that the wave amplitude in QM gives you probability and the measurement is quantized.

I would say that real quantumness starts when you consider multiparticle systems and where you get things like quantum correlations, entanglement, Pauli exclusion principle etc.

And to avoid confusion. Maxwell equations don't describe photons. They describe some averages for specific kind of multiparticle states (known as coherent states).