It is a natural consequence of the wave-particle duality. If we go back to classical mechanics, angular momentum is present when something moves along a circular path. Very small particles are also waves, and the frequency (number of times the wave function goes up and down per length unit) is a measure for their (angular) momentum. The wave function must now "fit" the circumference - meaning that if you go around the circle the whole 360°, you must end up with the same value of the wave function and are not allowed to have a sudden step. This only works if the circumference is an integer multiple of the wavelength. As a consequence, only certain wavelenghts and frequencies are allowed, and the same is true for angular momentum.

This is a very simplified picture, of course, but I hope it gets the principle across.

> Is the fact that angular momentum must be quantized a postulate of QM or is it derived from something more fundamental?

Quantisation of angular momentum follows from the mathematical definitions of wave functions and operators in QM. Specifically, it comes from the structure of the group of rotations in 3D, SO(3). In the end, this is Lie group and Lie algebra representation theory.

Angular momentum quantization isn’t a postulate, it shows up when solving Schrodinger’s equation for the hydrogen atom.

Quantization is a consequence of what happens when waves are bounded. In particular, quantization of energy, and therefore rest mass. Since all of the particles modeled by QM or QFT are done so in terms of wave equations, quantization of the solutions is a natural consequence.