By "pure black," I guess you and they are referring to a completely neutral black. Of the three things you ask about, only one potentially has that quality: darkness. "Black" animal fur and natural (not dyed) human hair are actually very dark brown, as a close examination will tell you. (I don't necessarily recommend close examination of a panther, except through a good 400mm or longer telephoto lens. At the very least, don't try it at home.)
Neutral black occurs naturally in charcoal, soot (once known as "lampblack" and commonly used as a pigment), graphite, some types of coal, certain types of marble, granite and basalt, and probably many other minerals I'm not immediately thinking of. What your art teachers are referring to, I think, is living nature, specifically plants and mammals. I've not delved into aquatic or insect life as regards color, so I can't say categorically there is no neutral black in either of those, but I would be surprised if there's a genuinely charcoal/soot black.
So yes, if your design includes a natural, non-mineral element that is supposed to look natural, avoid a totally neutral black, otherwise you will break the illusion. Very few designs would fall into that category, it seems to me, and not that many illustrations.
In designing for print, a "pure" black would be what's called a "built," or "rich" black, composed of 100% coverage of black ink overprinted on carefully-selected percentages of cyan, magenta and yellow. "Carefully selected" means the exact percentages required would depend upon the type of paper and the particular ink manufacturer. Black ink, even at 100% coverage, is a very dark gray on white paper, not black, and often is not entirely neutral.