Is it really illegal to damage money?

Yes, it is illegal to deface money. The U.S. government will replace worn out or damaged money if most of it is still identifiable. Two-fifths will earn the bearer half the face value; less than that gets nothing.

There wasn't a coin one could "deface" until the 100th anniversary of Abraham Lincoln's birth. The  Lincoln penny was the first U.S. portrait coin. 
Every American coin made since the Revolutionary War has some symbol of liberty on it.