* The specific edition: Different publishers might include additional characters or omit some for their version.
* The definition of "character": Do we count only named characters who have dialogue? Or do we include unnamed characters like servants or passersby?
However, we can say that "A Case of Identity" is generally considered a short story with a limited cast of characters. The story focuses on a few key individuals, primarily Sherlock Holmes, Dr. Watson, and the female protagonist, Mary Sutherland.
If you want a more specific answer, you could specify the edition of the story you're interested in.