Green’s Dictionary of Slang

1. (US) a little sister or young girl.

1941

J.G. Rothenberg ‘Peanuts! The Pickle Dealers’ in AS XVI:3 Oct. 187: Moose. A sister, notably a kid-sister who informs her mother when her brother is playing stick-ball, shooting dice, etc.

2. (US) a girl friend.

1974

(con. 1940s) E. Thompson Tattoo (1977) 228: Don’t wanta make me look bad to my moose.

3. (US) a young Japanese or Korean woman, esp. a prostitute, or the wife or mistress of a serviceman stationed in Japan or Korea.

[1868		S.P. Boyer diary 8 Apr. in Barnes Naval Surgeon (1963) 31: I am a young bachelor doing business in the place and I wish a ‘moosmie’ [musume — daughter, girl or young woman] as the young ladies are called. [...] I hire her by the month. [Ibid.] 13 June 54: John is rather fond of the Japanese moosmies and when ever he is on shore makes love to them].
1952		W. Sheldon Troubling of a Star 247: Sachiko. My moose. Always yapping for more dough.
1966		True Apr. 39: Each of America’s past wars has produced synonyms for the word ‘girl,’ which is possibly the most important word in the GI vocabulary [...] in Japan there was ‘moose’ (derived from the Japanese ‘musume,’ meaning girl or daughter). Since Korea was close to Japan, Korean girls also become ‘mooses’.
1972		R. Glasser 365 Days 230: Even his moose was beginning to annoy him.
1989		H. Rawson Dict. of Invective (1991) 256: moose. A prostitute, mistress, or girlfriend; military use since the occupation of Japan following World War II.