It means the dictionary definition, the common meaning of the word, as opposed to a specialized technical meaning or a poetic or not-literal meaning.
I saw an example not long ago where a writer wanted to praise people who work hard, take care of their families, are honest, and so on. So he said, "Mr Miller is not a hero in the ordinary sense of the word. But he is a hero in his community" etc. That is, he's not a hero in the normal sense of braving fighting the enemy in war or some such. But he's a hero because people like him make society work.
In this case, I can't say for sure without reading the context. But obviously a character in a story is not literally a real person. They're not real "in the ordinary sense of the word". I'd guess the writer goes on to say that they can be "real" in the sense that they can behave like real people behave, or that they can seam real to the reader.