How does Kaito Kid figure out the true identity in the 3rd movie of Detective Conan, "The Last Wizard of the Century"? I googled this and find that maybe when Kaito was disguising as a police officer, he could hear Conan talk to Agasa who calls him repeatedly Shinichi but I am not convinced
Kid just stumbled the phone conversation accidentally Conan had with Agasa. And there he finds the phone conversation pretty weird for a kid. The way Conan talks on the phone is not like an elementary grader. And confirms it even further from the way he deduce things from the murder case and why he knows and understands high-level infos - Kid got it all with just few clues.
You are looking for Boy Shinichi's Adventure Chapters 570-573, Episodes 472-473.