I will speak only for Russia given how different the situation would have been depending on where you were, but in general an urban carriage ride was very attainable for even the lower classes. Russian aristocracy, if they had the means, certainly preffered to own their carriage rather than rent it.

But just like today there are tiers of rented transportation (shared transit vs a private sedan vs a nicer sedan vs a limousine), several standards of carriage drivers existed in Russia: everything from peasants using transportation carts to a racing droshky to a bourgeois, intercity dormez was hireable.

This excerpt from the journalist Vladimir Gilyarovsky offers a tantalizing glimpse at the perceptions of such drivers:

From nine in the evening, cabbies began to gather from everywhere one by one, stood in a line on both sides of the alley... since the left was lousy with reckless drivers and golubchikami who paid the city for this exchange in large sums. Vanki, yellow-eyed pogonyalki - these cabbies of the lower classes, as well as kashniki [peasants?] who came to the capital only for the winter, gave khalturu to the police.

Some of the language is a bit opaque and rather rude, but we have the scene: some drivers pay for preferred access to clients, while the vanki (generally peasants using horses ill-trained and ill-equipped for the urban ride) are explicity for less well-to-do Muscovites.

Earlier in the collection Moscow and Muscovites, set at his arrival to Moscow in 1873, Gilyarovsky travels from Lefortovo to Khamovniki--a distance of 10km or so--for 12 kopecks. A skilled laborer could expect to make 6-10 rubles a month, or 600 kopecks, so this rate is very attainable.

In Misery, an 1886 short story by Anton Chekhov, a group of drunks demand to go from Vyborgskaya to the Green Bridge for twenty kopecks. This is 6-7km, and the narrator explicitly states this is not a fair price. After taking on only a few other fares that evening, he reflects that

I have not even earned enough to pay for the oats...

so we can imagine he expected to take on several further fares--in this case, he can not handle the misery of having lost his son to illness when none of his passengers care, before confiding in his horse. (As an aside, real wages fell after 1861 before bottoming out in the 1880s, so the quality of life for an urban Russian probably declined over this period).

The large, polished carriages with fine windows you might be thinking of were expensive but, for the middle classes (around ~25% of Petersburg inhabitants at the turn of the century) an option, while most residents could afford some form of transportation. Carriages play a prominent role in Russian literature becasue they were an important part of Russian life.

Mironov, Boris N. “Wages and Prices in Imperial Russia, 1703-1913.” The Russian Review 69, no. 1 (2010): 47–72.