coined 1899 by German chemist Heinrich Dreser (1860-1924) in German as a trademark name, from Latin Spiraea (ulmaria) "meadow-sweet," the plant in whose flowers or leaves the processed acid in the medicine is found naturally, + common chemical ending -in (see -ine (2)). The initial -a- is to acknowledge acetylation; Dreser said the word was a contraction of acetylierte spirsäure, the German name of the acid, which now is obsolete, replaced by salicylic acid.
Commercial names for medicinal products began in Germany in the late 19th century, when nascent pharmaceutical firms were discovering medical uses for common, easily made chemicals. To discourage competitors they'd market the substance under a short trademarked name a doctor could remember, rather than the long chemical compound word. German law required prescriptions to be filled exactly as written.