The “-ette” suffix can denote either a smaller version of a noun (e.g., kitchenette) or a feminine version (e.g., suffragette, usherette). Why is the feminine noun equated with being smaller? Sexism rears its head.
So what makes the TV tray in the photo a “trayette”? It looks neither small nor particularly feminine, though a woman is using it. I’m guessing that the marketers thought a feminine-sounding name might have greater appeal to female consumers of the era. Just a theory.