I bought enamel paint to paint a ceramic mug, but the paint can still be scratched off, why does this happen?

The original colour and finish were baked on in a high temperature kiln. What you probably need to do is apply a base/primer first, then the paint, and then a sealer coat. As for keeping it from flaking off, it helps if you wash it by hand in mild soap, with no abrasions, no chemicals, and in warm water as opposed to hot.