The Freud-Marlé-Leip story is not simple, but this seems to be the gist of it.
Lilly was the daughter of Marie (Mitzi) and Moritz Freud (a distant relative of Siegmund Freud, the psychoanalysist). Marie, her mother, was a sister of Siegmund Freud. At one time Lilly was a girlfriend of Hans Leip (he was handsome, intelligent, and attractive to women).

Leip wrote the song in 1915. Lilly married Marlé, a thespian, in 1918, thus did not have the name until then. The story thus seems false.

Lilly became an actress after marriage to Marlé. Formally, she used the name Lilly Marlé-Freud (a letter to her from the writer Rainer Marie Rilke, so addressed, recently sold at auction).

Anton Freud, a relative of Sigmund, told the story of Leip and Lilly, suggesting that 'Lili Marleen' was Lilly. Others in the family took it up and Lilly said it was true- it may have been a joke among friends and families because the Freud's (both Siegmund's and Moritz's) families and Leip were friends. Leip was not(?) Jewish and Siegmund professed atheism; his wife Martha was the daughter of a rabbi.

Between 1915 and 1939, the poem underwent changes. The original title was "Das Mädchen unter der Laterne." Leip added verse five in 1937, and used the title "Das Lied eines jungen Soldaten auf der Wacht."
Lale Andersen and Norbert Schultze(?) may have first applied the name "Lili Marleen" when he wrote the music in 1938 and she recorded it in 1939.

The story about the two girls Lili and Marleen also may have been fiction. I doubt that we will ever know for certain.
Leip was a story teller, writing novels and children's books in addition to poetry. He was a filmographer, working on films including "Judgement at Nuremburg." He also wrote non-fiction, his story of the Gulf Stream was translated into English and sold well. A minor artist, he made prints; one of two dancers recently sold for $550.

-Katja Behling, "Martha Freud, a Biography." Portions on line at books.google.com/books?isbn0745633382..
-Leip biography- "Lili Marleen," from Wikipedia, copied in http://www.radioswissclassic.ch/cgi-bin/pip/html.cgi?lang=en&m=entity&v=b&w=Lili+Marleen
-Freud biography- http://www.salus-institut.de/ siegmundfreudzentrum/awfreud.htm