Surge protectors are banned on cruise ships, and so are confiscated during boarding. Extension cords and cube taps etc. are discouraged, but not banned.
I'm asking specifically about surge protector incompatibility, and would like an answer that includes the technical details of the ship's cabin power supply.
I've asked in various places (not StackExchange) and have never been able to get a technical answer as to how a surge protector can cause damage.
Most responses are along the lines of "they cause fires", "fire is really bad on a ship", and "they ban all extension cords", so please don't give a similar answer.
To discourage "they ban all devices" responses, please look at this ad from Amazon and note the "Non Surge Protection & Ship Approved":
(Note that I wrote this last night before the answer from @StainlessSteelRat)
Learning that ships use an isolated-neutral system was a great help.
Here is a summary (many details omitted) of where I am so far. In particular, I run into what seems like dead ends (in bold) where the bad effects aren't nearly as bad as they at first sound.
Wiring Systems:
Normal house wiring (North American standards) is asymmetric, having one line at 120V, and the other shorted to ground at the main breaker panel. The neutral side (larger slot in outlet) is, in theory, safe to insert a fork without shock. The live side is, in practice, dangerous to insert a fork.
Ship wiring is symmetric, having two lines, each at 60V and 180 degrees out of phase. Both sides in theory pass the fork test (similar to birds safely sitting on a distribution wire).
Surge Protection:
With normal house wiring, when a surge protector notices a voltage spike between ground and the live wire, it diverts the excess current.
a surge protector has a continuous but small current (for the detection), and occasionally a brief but large current (when clamping).
Surge Protectors on Isolated-neutral systems:
The connection between one of the live wires and ground causes the system to behave like a grounded system. The current is normally small, but with many cabins on the same circuit, it might become significant. This makes the system more dangerous to users, but no more dangerous than home wiring.
The grounding occurs on the side of the circuit that connects to the live socket. Devices that rely on the difference between the two sides will be compromised. For instance, the on/off switch is normally on the live side, but would now be on the neutral side, leaving voltage on the device (e.g. even when switched off, the threaded part of a light-bulb socket would be live). This makes the system more dangerous to users, but no more dangerous than home wiring that has live and neutral reversed.
There will be multiple connections to ground in the circuit, which can create ground-loop currents. This can be annoying (e.g. hum in sound systems), but doesn't create any danger.
Speculation:
If the two wires aren't different colours, or if the installer knows that it doesn't make any difference, one line could end up connected to the neutral outlet in one cabin and to the live outlet in another. That could increase the continuous current leakage through ground. In the case of an actual spike, both sides would be briefly clamped to ground creating a large current and possibly tripping the circuit breaker. This would be annoying, but not dangerous.
I'm sure that the ship's electrical system has devices to monitor and regulate the power supply. The current leak to ground caused by surge protectors might cause this mechanism to get false readings. This could be dangerous to users.
There is an effect on metals known as stray current corrosion. Even without any added electricity, random currents occur in metals, and, particularly in the presence of salt, will cause the metal to corrode at points that act as anodes. Many marine vessels have magnesium-anode devices attached to their hulls to mitigate this effect. I imagine this is the reason that ships use isolated-neutral wiring, to avoid introducing any electrical currents into their hulls. This isn't immediately dangerous, but would be immensely expensive in the long run.