There are no luxury train services in the United States.
The two long-distance services, Amtrak and the Alaska Railroad, do offer premium services on their regular routes with greater comfort and privacy and a higher level of service than coach. You can even buy multiple bedrooms on Amtrak and have them merged ensuite for more space, or buy out a whole car, as sports teams or other groups sometimes do. I would recommend taking a sleeper car trip if one ever has the opportunity, and the service is very friendly; just understand that even by American standards, nothing about the experience would be mistaken for "luxury."
The way luxury rail travel is done in the U.S. is on private rail cars, which are attached to one of the aforementioned operators' trains. This is more comparable to chartering a yacht than it is to booking an ultra-premium airline seat, but is common enough that Amtrak provides guidelines for privately-owned railcars, and there is an American Association of Private Railroad Car Owners. AAPRCO provides a list of travel opportunities for members of the public (often for promotions or repositioning), but for the luxury experience you'll likely need to go through their directory of cars available for charter and contact the owner directly to negotiate a booking.
There are any number of operators, like Los Angeles-based L.A. Rail, that do rent out cars to the public. Seattle-based Puget Sound Private Car is another, and Pittsburgh-based Berlin Sleeping Car another. I have never used any of these companies and do not endorse them, but I cannot find any centralized portal to them, and there are likely to be many more in different parts of the country. Berlin suggests a range of $2500-7000 per day; like a yacht charter, the circumstances and services of each trip are unique and will be priced accordingly.