The phrase King of the Jews is similar to the Jewish title "King of Israel" in 15:32.
You have said so: The response Jesus gives to Pilate's question appears in all four Gospels. Jesus clearly understood himself to be a king (10:37–38; 11:9–10; 12:35–37). Pilate eventually crucified him for this reason (15:26). But Pilate knew Jesus was not a political threat (John 18:33–39).