Guess what? You can carry your passport off the ship, your drivers license or leave everything on the ship. It's your problem if you can't get back on.
If you leave the ship you are being irresponsible if you don't carry some sort of ID. If you are in the Caribbean or in most European countries you won't need to carry your passport with you but you better have a driver's license or other government issued ID. Sometimes it's required to get back on the ship and sometimes it isn't. It can vary. I've had to show it in Cozumel but other times I have not.
You can take a photocopy of your passport on shore and in the event you miss the ship it will help you get a temporary passport or clearance to fly home. The passport number is the primary bit of information on that piece of paper but in reality that's all it is, a piece of paper with your smiling face on it. It is worthless as identification or proof of citizenship.
I will leave my passport on the ship or with the ship when they require you to turn over your passport. However, if I am going on a private tour that has a chance that I may not make it back to the ship (overnight in Cairo, Beijing or even Rome) then the passport goes with me. I will not be stuck in a foreign country without it. If I want to get to the next port or get home I don't want to deal with the delays I will encounter with my "photocopy". If I missed the ship I probably would have dealt with enough already.
I have told the tale of our 2006 China cruise where the ship's overnight excursion in Beijing was held up for almost four hours because the hotel would not accept photocopies of the passports and a car had to drive two hours back to the ship to retrieve the passports. The guides would not continue the tour until everyone was checked into the hotel so the people had to sit and wait.
I booked a private overnight tour and had to go to the Hotel Director before they gave me my passport, telling me the whole time that all I needed was the photocopy. I had dinner with the Hotel Director the night after we left Xingang (Beijing) and he apologized for telling me this when HAL had to do partial refunds of a very expensive tour to people because they told them all they needed was a photocopy.
So: Take whatever ID you want ashore and if some official asks for it I hope you have the right one.