The "studio" got it's name from the space of the artist's studio.
(Oxford English Dictionary)
The work-room of a sculptor or painter; also, that of a photographer.
1819 Edin. Rev. XXXII. 322 The greatest work which proceeded from his [Cimabue's] studio, was his scholar Giotto.
studio apartment U.S. = studio flat, a flat containing a spacious room with large windows, which is or resembles an artist's studio; more recently, a small one-roomed flat;
1903 Archit. Rec. July 240 The most economical way of combining a good high studio with an economical disposition of space would be to make the studio apartment two-storied in the service and living portions, and only one-storied in the space devoted to the studio. 1929 Washington Post 1 Sept. 6/5 Studio Apartment $100.
3 December 1884, New York Times, pg. 1:
The Demorest Studio Building, at Nos. 4 and 6 West Fourteenth-street, took fire early last evening ,and the crowd that packed the street in a minute witnessed some narrow escapes.
They were Miss Donlevy, Recording Secretary of the Art Association, who was in its parlors on the third floor, and Miss Mary Ward, of Texas, who was in her studio apartment.
9 December 1886, Life, pg. 360:
And later, when he had taken the new studio apartment where they were to live after the honeymoon, she said to him: "I honestly thought those apparitions were real."
26 November 1896, New York Times, pg. 8:
"'Trilby' has given people a wrong impression of studio life," says the wife of a New-York artist, who makes her home, with her husband and children, in studio apartments. "Life in a studio is as regular as in a private house. People who make their homes in studios live as other people do who try to solve the problem of living in New-York by doing light housekeeping."
8 April 1897, New York Times, pg. 19:
Hereafter the class will meet at Mrs. A. D. McCredy's studio apartments, in the Carnegie Building, at Fifty-seventh Street and Seventh Avenue.