Yes, it’s “correct”.
(And by the way, slang does not mean “incorrect”. It’s a specific sort of casual vocabulary or expression, not incorrect grammar.)
He showed me myself.
Here me is the indirect object and myself the direct object. It wouldn’t sound right to say to use me for both direct and indirect or prepositional as in "He showed me me" or "He showed me to me", so one of those two got swapped to myself.
It’s not quite true that you can only use a -self pronoun when the subject and object refer to the same person or thing. For one thing, the -self forms are also used emphatically, as in "I myself called the sheriff."
So you could think of this as a form of emphasis to distinguish the two pronouns so it doesn’t sound clunky. All these variants would be understandable in a way that the doubled me version would not:
- He showed me myself.
- He showed me to myself.
- He showed myself to me.
- He showed me my own self.
- He showed me to my own self.
Those all pretty much the same thing.