You are required to
have a passport to travel internationally. You are not required to carry a passport on your person at all times while in the foreign country.
Seriously? Why on earth would you be detained by authorities on a
DCL excursion? I don’t believe any disney excursions currently involve crime sprees or international espionage. And if you are stopped by “authorities” for whatever reason, they can’t arrest you just for not having a passport as foreigners do not to carry passports at all times. In the remote Agatha-Christie-Novel scenario where you are actually arrested on suspicion of a crime while on a cruise ship excursion, you will obviously be missing the ship’s departure that day so the crew will retrieve your passport and give it to the ship’s agent at the port, where you can arrange for it to be retrieved pending your criminal trial. But again, seriously?!?!?
Do you also carry a two week supply of food and clean water with you on excursions? That is also on the webpage you linked, because it is about what to have in the event of a major crisis in a country you are visiting.