Does "stomach" itself have a meaning of "gut"? Or "stomach" becomes the meaning of "gut" only when it is used as this sentence?
Yes, to "not have the stomach" for something means to not have the "guts," determination, or courage to do something. The speaker has used this expression to state that Draco Malfoy is cowardly and undetermined, just like his father. Similar expressions are "he doesn't have the guts" and "doesn't have the nerve." Ex: He doesn't have the guts to take the risk. Ex: He doesn't have the nerve to do anything risky.