How does martial arts help kids build confidence?

Quick answer: Confidence here means repeating a hard kick in front of a coach, not a slogan.

They get a correction, try again, and the class sees the improvement. That loop is the point.

Is taekwondo or jujitsu better for shy kids?

Start Taekwondo. Closer grappling can wait.

Ask the school. Some hours want parents in a viewing area so the child listens to the coach.