That’s not easy to say, since upside down tarts and cakes have been made since at least the day of the great Antoine Carême. Indeed Carême published recipes for several glazed gâteaux renversés, as he called them, in his Royal Parisian Pastry Chef cookbook, which is dated to the 1830’s.
So the idea is not very new and indeed probably came to prominence in the 1880’s with the rise of tarte Tatin, the famous upside-down apple tart made from sliced apples, caramel and puff pastry. The dish was “invented” at the Tatin Hotel in Lamotte-Beuvron, but its fame soon spread and was being made everywhere in Paris by the turn of the century.
From there there’s no bright line to modern upside-down cakes, save to say that one particular variant, pineapple upside-down cake, was all the rage in the States in the 1960’s, just after Hawaii was made a state. That iteration is still the definitive upside-down cake in America, though they’re being made with all manner of fruits these days. Thankfully, they never seem to go out of style.