The part about them being too closely related to Humans sounds like BS, but there is a mushroom that is perfectly safe the first few times you eat it, and then eventually makes your immune system attack your blood cells.
There's a Paul Stamets video where he talks about how mushrooms are so closely related to humans that we both fight off similar pathogens and that is why they are so useful to us for medicine (penicillin for example.)
In the Paul Stamets TED talk, he never says that humans specifically are genetically close to fungi. He said that between all the different kingdoms of life, animals and fungi were more biologically similar than any other two kingdoms.
That definitely explains why we can borrow useful defenses from fungi, like antibiotics, but it's definitely not a reason to believe that our immune systems would have any difficulties differentiating between certain fungi and our own bodies, at least not for reasons related to direct genetic similarities.
Yeah the similarities make sense when you look at sponges and sea lilies and the like, but the difference between a mushroom and a mammal is incredibly vast