As you have pointed out, it's usually worded "I love myself" instead of "I love me". In fact, almost always. You ought to use the reflexive in such a case.
However, this is not a mistake. Quite the contrary. It's a humorous situation, somewhat in the same spirit as referring to oneself "in the third-person".
"because Charlie loves himself, that's why".
Anyway, if Charlie had said "myself" it would have been too correct, too serious, and therefore not right for the line*.
( * A "line" in a television show or movie is what's being spoken by an actor. Definition: "A sentence of dialogue, especially in a play, movie or the like.")
Is it a common mistake in daily conversation?
No, these words are rarely mixed up by mistake.