An anemone is not considered a coral, however...mushrooms are also called coral anemones so that could cause some confusion.
Mushrooms can reproduce several ways. By shifting over and leaving a bit of their stalk/foot on the rockso that the leftover stalk creates a new mushroom; by stretching apart until it becomes 2 separate mushrooms, etc.
We can also propogate them ourselves by fragging. Fragging sometimes mimics what I just described above by either slicing through the stalk horizontally, or by cutting through the crown in pie shaped wedges. I've heard tales of people putting them in a blender and pouring the puree into a tank. I'd not try that last one personally.