This term is often assumed to derive fromMickey Mouse,the reference being to a bottle of small size. However, Disney's character did not appear till 1928, and the term is shown below to have been in use before then.
1927  "We takes a chase round a block and gets a mickey of de demon, which put us on our way rejoicing. . . ."
1950  An American pint holds 16 ounces, a Canadian "mickey," 12 ounces of rye, or 13 ounces of Scotch.
1962  But even then very few of the electorate went reeling to the polls with $3--the price of their votes--in one hip pocket and a mickey of rye in the other.
1966  Bradley said the four . . . had gone through two mickeys of whiskey.