The answer is yes. Also the quality and style of the carriage and horse(s) (the entire thing being called an equipage) were a show of wealth. The styles and types of carriages were many, and the quality of them also ran the gamut (similar to the difference between owning a Toyota Camry and a Rolls-Royce limosine). You could have a simple carriage ordered from the sears catalog and a former plow horse (your Toyota Camry) or you could have a formal Landau drawn by four well matched fine horses (Rolls-Royce Limosine) Carriages for hire were very common, as you could easily hire a carriage to take you to the opera or to dinner for the evening, etc. Owning your own carriages entailed that you had your own stable, your own servants to care for the horses and the carriages, and could afford all the upkeep required, which for someone in the city meant that you could command great wealth.

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Riding horses in the city were not very useful, as walking or taking carriages or early public transit was the easiest way of getting around. Owning a horse in the city solely to ride for pleasure would have been a sign of wealth. Rotton Row in London was a place where the wealthy of London would ride and drive just to see and be seen and enjoy a nice day out in Hyde Park. This was not a past time available to anyone who wasn't in the 1%.