Sure, carriages and stagecoaches existed. Rickshaws too for that matter, depending where you were in the world.

We also called them carriages, stagecoaches, and rickshaws, not taxis.

There were also handsom cabs, but I don’t know if they used taximeters before they became electric in 1897, as they are not referred to as ‘taxis’ until 1897. The modern taximeter was invented in 1891, but the first taxicab to be equipped with it was in 1897 (and it was gas powered, not electric).

The ‘taxi’ part in taxicab doesn’t refer to driving someone around, btw. It refers to the meter used to derive how much you pay. Nowadays, we would use it to to refer to someone driving you around though, because that’s how language changes.