NYC: Are the horse-taxis that tourists ride in left over from the era of the horse?

We’ve all seen how tourists in New York City take carriage rides in Central Park, a thing which I’m assuming no natives do. Assuming they still do this, what is the history of the organization that provides this service? Were they the original taxi system of the city, and found their clientele limited to sightseers, once motor taxis came in? Or did somebody come up with the idea of horse-drawn carriages for tourists later on?

I found a little here and here