It's not a misinterpretation, and you're missing a key reason as to why:
Killian's parallel story arc to Tony's, and the whole damn theme of the movie. At the end of the film, Tony Stark basically says to the camera that Iron Man isn't defined by the suit and the gadgets, that ultimately he is defined by who Tony is as a person. He says at the end "I am Iron Man," meaning that with or without the suit, thats who he is as defined by the core of his character. Just like how The Mandarin isn't defined by the rings and the robes and the funny accent, he's defined by who he is, and as Killian says, at the core of his character he is The Mandarin.
Killian fills The Mandarin's role in the story and shares most of the same character beats, motivation, MO, and general personality. Even though he lacks the surface resemblance (until the shirt comes off and we see the dragon tattoos anyway, another clue as to him genuinely being The Mandarin), the film's own theme is that it's what's inside that counts, and deep down Killian is The Mandarin. He's certainly much more like the actual character of The Mandarin than some immortal Chines sorcerer like people keep suggesting.