As is so often the case, fact is more interesting than fiction. The other woman during Stravinsky's years in Paris was Vera de Bosset, not Coco Channel. The two engaged in an illicit affair while he produced short compositions for the French piano manufacturer Pleyel. Eventually, Vera would become a major part of Igor's life, traveling with him throughout the 20s and 30s, with the two eventually getting married after Stravinsky's first wife, Katerina, died in 1939. They settled in the United States.
Of course, one can still speculate that Igor and Coco may have been more than casual acquaintances in Paris, but Vera was a woman closer to Igor's heart and soul. One who did understand his music, and wasn't just a "shopkeeper" ; )
The portrait is by Sergei Soudeikin, depicting Vera as she would have appeared in 1920. You can view the painting at the Fine Arts Museum of San Francisco.