That is absolutely not true... least not anymore. Probably not since WWII. Larger ships, ie: cruise ships, cargo, freighters, tankers, etc. have all had fresh water-makers in my boating lifetime. While it is true that there is special soap for salt water, almost no one uses it anymore (sailors on small sailboats without a water-maker - being the exception).
Believe it or not, it was in 1791 that (of all people) our future President Thomas Jefferson, described the results of a simple desalination process (taking the salt out of salt water). The information was to be printed on the back of all the papers on board ships so that there would be a source of fresh water in an emergency.
The first large capacity fresh water conversion units were made so that steam ships could have fresh boiler water. It was this process that made the steam ships capable of ocean travel, and made it so economical. Since steam engines (and thus steam ships) could not run on salt water, their cargo bays (or space) would otherwise have been full of fresh water to run the ship.
By World War II - hundreds of smaller mobile desalination units were in use and all military vessels had them.
In the US, there was a program established by the Secretary of the Interior called OSW (Office of Saline Water). They set up the Maritime Saline Water Act of 1952. which out of fear of thirst at sea, and fear of having unclean, unhealthy crew - established a Maritime law requiring fresh water makers with enough capacity to furnish all passengers and crew amply fresh bathing and drinking water on board all ships traveling to and from all US Ports.
So... to answer your question, No, not on large ships, least not since 1952. And yes, submarines had fresh water showers and water to drink in WW II.
Happy & Safe Boating,
John