I believe what your teachers are referring to is the idea of the 'blackest black.' Only in recent years have we achieved even synthesizing 'true' black, black that consumes so much light, that it creates the appearance of the absence of all light, the illusion of a flat, dark void, a paint called Vantablack. There is nothing like this in nature. While what you mention are fairly close, these things still reflect back some amount of light. Perhaps a room sealed off from all light is perfectly pitch black, but this isn't a tangible concept, and you have to admit, you would feel quite unnerved by this, right? This isn't a case of light being absorbed or reflected, but the complete lack of it. Even at night, our eyes can adjust, and take in the little bit of reflected light in the gloom.
The advice that using pure black appears synthetic is maybe slightly subjective, more advice than outright fact. I'm a fan of tempering my darkest blacks with a little color, coloring my lineart to help it feel cohesive, etc. I think the real warning behind these statements may be to use your darkest, purest blacks with intention, rather than the idea that you have to use pure black to convey darkness or contrast. Don't jump right to your most pigmented black or #000000/(0,0,0) right away, or if you do, consider how your design might benefit from tinting the black.