As in the other thread, this depends on the content of the course/textbook. If this is about Peano arithmetic, then x = y -> S x = S y holds not only for S but for any unary functional symbol. This is an instance of one of the axioms for equality (and Peano arithmetic is by definition a first-order theory with equality).

On the other hand, x # y -> S x # S y, which is equivalent to S x = S y -> x = y says that S is injective. Other functional symbols may be interpreted by non-injective functions. Injectivity of S is also one of the Peano axioms.