Okay, I know that this is probably Plot-Induced Stupidity, but... Hasn't Nakamori-keibu ever heard of Facial Recognition? Yes, Kid wears a top-hat and a big monocle, but 3/4ths of his face is still visible, and what with the fact that everybody and their GOLDFISH is there and taking pictures and videos of a heist, especially Kid, who's the FOCUS of the heist, you'd think they'd subpoena the media's tapes and run them through Facial Recognition systems for civilian identities, such as driver's licenses, school yearbooks (they usually have an online version), newspapers, SOMETHING.
I have no idea how Japanese law works in this case, but if it were America they couldn't do that. Such databases only exist for people already in police databases, checking things like driver's licenses and school yearbooks would be an invasion of privacy. Again, I'm not sure if that is true in Japan, and even if it is it wouldn't apply to Shinichi, so he's got no excuse there.
Well, I'm not sure how ubiquitous that technology was back when Magic Kaitou was first released, but there was an arc where Nakamori saw KID's un"masked" face, and started investigating his civilian form. KID's response was to fake being on a date with Nakamori's daughter at the same time as he committed another heist where his identity is revealed again, this time as the aforementioned daughter, reminding Nakamori that not only is one of KID's primary skills is his insane mastery of Latex Perfection, but he also just really enjoys screwing with Nakamori's mind for giggles, so there's no real reason to assume that his face really looks like that.
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This has actually been officially tackled in the Magic Kaito manga; Nakamori gets a good look at Kid and recognises his daughter's best friend. Aoko subsequently tearfully begs Kaito to take her on a date the same day as an announced heist, to be his alibi. Hilarity Ensues. In the end Nakamori makes a grab for Kid, who's face tears off to reveal Aoko's face and hair underneath, and Kid openly mocks him, saying he knows ALL the people who are close to Nakamori, and may possibly appear as the inspector himself next time. In the end Nakamori decides, legitimately, that he'd been duped and that Kid never appears with his real face at all, which is helped by Aoko's testimony of where Kaito was all the time.
I'd very much like to know what kind of show Toichi was planning 18 years ago that required the use of a cape that turns into a hang glider.