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.

TIL Stamets is named after a real mycologist.