The Jews accused Jesus of lying. Pilot didn’t care what Jewish law he broke, really. He just wanted to appease the jews because he didn’t want trouble. He could only punish him for breaking Roman laws. When Jesus admitted he’s the king of the jews, that’s the law he broke as a crime against Caesar. Not lying.
Being king of the Jews wasn't a crime against Ceasar though. The Jews had kings while under Roman rule, didn't they?