The things that we call mushrooms aren't the actual organism. That's just fruiting body of the organism, analogous to a flower in a plant. Picking the mushroom doesn't kill the organism, since the mushroom itself is really only a very small fraction of the entire organism. The actual organism is actually underground. The organism is large network Berg-like microorganisms that fused together into a complex system of "roots" called hyphae.

Hyphae do not photosynthesize like plants do, they eat things in the soil for their energy. They do not have a cellulose cell wall like plants do, their cell wall is made of chitin (the stuff that bugs use for their exoskeleton). Genetically, they are (very slightly) closer to animals than they are to plants. Morphologically, they resemble protists than anything else. Chronologically, they evolved significantly after plants evolved, and they evolved from a proto-animal lineage. In some species, the microorganisms that make up the hyphae can decide to unfuse and start living on their own (at that point, we call them yeasts). How and when they decide to fuse/unfuse is unknown and it's a fairly large area of research, especially since that transition is often associated with their ability to cause diseases (yeast infections).

Mushrooms are the closest things we have to aliens, and the fact that we just eat them and think nothing more of it is genuinely amusing to me