You need to show your passport for Shanghai Disneyland but not for any other parks.
Although in France, Hong Kong and Japan, it is a legal requirement to carry valid ID so it's safer to your passport with you anyway. If you are at least 15 years old in Hong Kong or 16 years old in Japan and found without ID (I don't know the age for France), you commit a minor legal offense and possibly can be arrested. The US is actually weird as a developed country with no laws requiring ID.