It's the present in the US. Many people own personal train cars, and you just contract with Amtrak to hook you up and you're off on vacation. You can even bring Babu. You can rent personal cars as well, though you probably should make sure yuor ocelot is housebroken if you're taking a rental.
Now, I say "many" but what I means is that's more than a few. Many is still probably in the 3-4 digit number (I'm guessing). And you'd be correct in assuming that it's not a luxury most people can afford. But it does exist.
I was just googling around, and it looks to me like a private rail car costs something like a 2nd home, storage fees similar to property tax, $4/mile to have Amtrak haul you around. Basically a vacation home, but mobile. Definitely a 1% thing, but not billionaires-only. Probably way more prestige in saying you've got a private rail car than a beach house. At least among a certain segment.
DO they still? Last I heard Amtrak was no longer taking private train cars as too many were not in good mechanical shape and thus a large cause of their delayed trains.
Railroad suburbs exist! Streetcar suburbs as well. Was actually the norm outside of the city core until they started ripping up all the rail lines to build highways.