As I understand it, the value of Planck's constant h was initially introduced, in some sense, as a fitting parameter to understand blackbody radiation and the quantization of photons.
Pretty much. Max Planck introduced it to get past the ultraviolet catastrophe.
In that sense, one can think of it as simply the proportionality constant between the energy and frequency of a photon.
Don't. It's more than that. It's telling you something important about the nature of light.
However, we also have other relations such as the de-Broglie wave relation which use h, and some at a deeper theoretical level like in Schrodinger's wave equation, the commutation relation [x,p]=iħ, and the quantum path integral in the path integral formulation of quantum mechanics or quantum field theory.
Yes we do. Note that it's the de Broglie wave relation and the Schrodinger wave equation. Also note that the path integral formulation is to do with wave function.
Why is ħ which is essentially a proportionality constant related to light, so deeply tied to and seemingly universal in quantum theory?
Because quantum theory is the strange theory of light and matter.
To be concrete, why does the value of ħ for the photon have to be the same as that of the de-Broglie wave relation, and why is ħ the same for all massive particles?
Because of the wave nature of matter.
Why does it have to be the same as that in Schrodinger's wave equation or the weight in the path integral or in the commutation relation?
Because of the wave nature of matter. We make matter out of light in pair production. Then we can diffract electrons and other particles. Because in atomic orbitals electrons exist as standing waves. Then we can annihilate particles with their antiparticles and get the light back. See Newton's Opticks query 30: "Are not gross bodies and light convertible into one another?" Yes they are.
Why does classical theory generically stop being relevant on action scales close to ħ.
It doesn't. We have classical electromagnetism and optics.
Qualitatively speaking, it doesn't feel like quantum theory intrinsically restricts ħ to be the same in all these cases, in the same way that general relativity identifies inertial mass with gravitational mass. Is it simply black magic that all of these are related, or is there something deeper?
There's something deeper. See Leonard Susskind in this video. At 2 minutes 50 he rolls his marker round and round saying angular momentum is quantized. You doubtless know that a sine wave is associated with rotation. The quantum nature of light is to do with that: roll your marker round fast or slow, but roll it round the same circumference, because Planck's constant of action h is common to all photons regardless of wavelength. Photon momentum is p=h/λ. The momentum can vary, the wavelength can vary, but h doesn't. The dimensionality of action can be given as momentum x distance. And it's the same distance for all wavelengths. Ever played Spanish guitar? You vary the wavelength with your left hand and you pluck with your right. In most situations the amplitude of your pluck doesn't change. Like Maxwell said, light consists of transverse undulations. Take a look at some pictures of the electromagnetic spectrum. Notice how the depicted wave height is the same regardless of wavelength? Note how all those transverse undulations have the same amplitude? You are looking at the quantum nature of light, at that something deeper, and it is hidden in plain view: