Its possible to "frag" mushrooms. Its not as simple as slicing it up and placing it on a rock so it will attach however. Yes you can slice them up but you need something like an elastic band wrapped around the mushroom and rock to hold it in place for a week or so whilst it re-attaches.
The reason its done is to make them grow on faster. Some mushrooms take a very long time to multiply so this can be done to make the growth faster. Although they are animals, they as also a sort of living structure of signel celled organisms... (probably not correct but i dont know how to explain it batter). For example.. if you get your arm cut off it will die and you wont get another.. cut up a mushroom or any other coral will mean they should all regrow into new colonies.