We stumbled across one of these photos of a horse getting a shower in 1910, and it turns out that around that time the ASPCA brought not only these showers, but drinking fountains to the thousands of working horses in the city. Not only that, but EphemeralNY recently pointed out the SPCA's plan in 1902 to give free straw sombreros to the horses to shield their eyes from the sun! They were made by a horse outfitter in Union Square, and look just like regular hats (you can see one in one of the photos above).
We asked the ASPCA about all of this, and they referred us to Dr. Stephen Zawistowski's book, Heritage of Care, which states: "Ten elaborate water fountains were built in a variety of locations throughout the city... and hundreds of water troughs placed. In later years when fountains and water troughs had all been removed because horses for the most part had disappeared, the ASPCA carried water in a truck during the hot weather for the several hundred carriage horses and other novelty horses who remained."