A poor analogy, as it's the same as saying that I should expect a cow to grow mold because a steak will get moldly over time. That said, the conditions that allow many mushroom-bearing fungi to flourish are also ideal for mold growth, so it's not impossible to see mold on mushrooms. I've had to throw out my fair share of monotubs due to the growth of green mold

They can, but they tend to be more mold-resistant than comparable plant matter (and from what I've read some varietals won't mold, but I'm not an expert so I can't verify or deny that claim), assumedly because there is less viable material for it to eat. A house made of wood is vastly more condusive to mold growth than a house made of mushroom would be.