Only in the US could the most collectivistic and efficient mode of transport be perverted into yet another incredibly inefficient and individualistic toy for the wealthy. I can't seem to find anything like that anywhere else.
It anppears to be Amtrak’s greater flexibility and uniformity of gauges in North America that allows this. Europe has more of the historical private wealth that would still own and want to operate a private train or carriage.
It's probably more that distances were shorter, the crazy rich could afford an entire train, and the less-rich would use private luxury carriages owned by the railway companies.
Since the 1950s or so, the flexibility has been gradually lost as trains become mostly fixed formations for speed, safety etc, so that certainly explains why it doesn't exist now in Europe.