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whenever I heard of the "wave-particle" duality of photons, I always thought they existed as a weird combination of physical waves and particles.

Since finding about De Broglie wavelengths and matter waves, however, I have learnt that the "waves" in quantum mechanics are actually probability waves; the oscillations are actually of the real and complex components of the waves function and not of some physical quantity.

however, isn't the oscillating quantity for electromagnetic waves the electric and magnetic field. Are electromagnetic radiation "waves" of probability or real physical waves? Is the waves nature of photons fundamentally different than that of matter?

Qmechanic
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Photons are probability waves of photons. Photonic waves if you will. They are excitations of the photon quantum field, in the same way electrons are excitations of the electron quantum field.

However, in the classical limit, photons look like simply oscillations in an electric and magnetic field.

However, quantum field theory (qft) tells us that there actually is no "electric" or "magnetic" fields, but only the single photon field, which interacts with all other fields of charged particles.

Electromagnetism is simply the exchange of virtual photons between charged particles, and changes in this look like changes in an electric field that propagates through space.