Identity is not the only function for which that holds. As an immediate example, constant functions have this property as well. More interesting examples exist as well.

Also what was stated was not an equation for any old fixed point but instead idempotence—a specific kind of fixed point. In particular, this describes precisely the shape of a "validation" function so long as it either succeeds on all inputs or has a notion of failure in either its computational context or it's range ("span" I guess).