I think that they are referring to Paxillus involotus

It is quite an interesting mushroom. It was considered "safe to eat" for a long time, but it contains an antigen that a human's immune system can learn to attack.

The antigen is still of unknown structure but it stimulates the formation of IgG antibodies in the blood serum.

I once looked into whether this immune response builds up over many exposures, or if it is a random event that has a probability of happening for each exposure. I don't remember finding a convincing answer... If it is a random event, then mushroom could be considered a "Russian roulette" mushroom that will usually provide a nice meal, but, if unlucky, you may experience the following:

Poisoning symptoms are rapid in onset, consisting initially of vomiting, diarrhea, abdominal pain, and associated decreased blood volume. Shortly after these initial symptoms appear, hemolysis develops, resulting in reduced urine output, hemoglobin in the urine or outright absence of urine formation, and anemia. Hemolysis may lead to numerous complications including acute kidney injury, shock, acute respiratory failure, and disseminated intravascular coagulation. These complications can cause significant morbidity with fatalities having been reported.

I agree with you that this is probably unrelated to the "generally similar to humans" comment. I feel like this fantasy is a combination of the above fact mixed in with the fact that the Fungi belong to the Opisthokonts, which places them closer to animals than plants, and so they share some interesting cellular characteristics with us. This places them closer to animals than plants, but "generally similar to humans" is perhaps a bit of a stretch ^_^

But, it is just a meme about a guy being hyped about mushrooms. Hopefully people don't expect memes to be super accurate 😁

There are fungal infections that cannot be treated as there are no good targets for fungicides that don't also affect humans

That's just a long way of saying the same thing. Not as catchy imo.