There's a reference to this song in a Salt Lake City newspaper article from 1899 where a woman in the city jail was heard singing this song. It must have been well known for people to know the reference without mentioning the title of the Song.

After calming down that evening, “from her prison cell last night shrouded in gloom there came a plaintive melody as from a heartbroken mother. Soft and low the notes were, wafted through the iron bars into the grim street below and all who heard bowed reverently as they tarried on their way to listen.” She sang: She’s the only girl I love. She’s got a face like a horse and buggy. She’s the only girl I love. Oh fireman save my child!”
Salt Lake Herald 30 May 1899