I find 2 Martin Lee's in 1910 census. One is counted aboard the USS Virginia stationed in Hampton Roads VA, has married 4 yrs and Immigrated 1888 and is an listed as sailor. The other Martin Lee is b. 1869 (age 41) is listed with family in Manhattan, NY married 6 yrs has Oiler-U.S.Navy, Immigrated 1886. This Martin is in the 1920 census still with family in Manhattan and this time he is listed as an Engineer on Steamship and no reference to still being in Navy.
There is a Martin Lee in the U.S. Navy in 1900 census on the ship USS Albany ported in England.
So it looks like he was in the Navy by 1900 to at least 1910, then by the 1920 census he is out of Navy and on a Steam ship.
On Ancestry in the Oregon Passenger and crew lists.is a Martin Lee age 45 (b abt 1872) a wiper aboard the vessel "War Monarch" arrived in Astoria, Oregon July 19, 1917 from the port of San Francisco CA.Shipped 1/13/1917 in SF.
Have you tried the NARA for his Navy Service record? www.archives.gov
There are passenger lists and ships lists on the LDS FamilySearch.org website and even more on microfilm and in book form at a Family History Center.
Unless you heard the directly from Martin his employment and sailing jobs,then all such "family lore" is just that...lore.... a knowledge or belief that is not necessarily accurate.