Well, yes and no. During my time running security at the Norfolk Naval base, they had a temporary display at the base museum, a collection of Dixie cups thru the ages.
No one is quite sure where or when the Dixie Cup originated, though it seems to point to the first ships of the ABCD steel fleet. The first mention in regulations is 1886.
The originals from the 1880s were several pieces of pie shaped sail canvas, sewn to make a mini sou'wester, done by ships sailmakers. Almost immediately they began adding stitching around the edges, for stiffening, and because the wearers liked to roll the edges up, to catch cinders. Or pull the front down in a fedora pattern. Sometime in the 1890s some began shortening the pie wedges and adding a narrow circular rim. Over the years this would grow.
However, this was not a standardized pattern, early regulations are rather vague, it varied from ship to ship, or sailmaker to sailmaker. Ships captains might insist on a particular style, but navy wide it was all over the place. By the beginning of WW1, you could find patterns that are nearly identical to modern caps, and some that go right back to the 1880s originals.
The massive fleet expansion plan of 1916 would provide the impetus to finally, fully standardize the dixie cup, with the final pattern being fixed in the early 1920s.