The Rail Passenger Service Act of 1970, which established Amtrak, specifically states that, "The Corporation will not be an agency or establishment of the United States Government"
But indeed it is a special corporation that is not quite private, yet seeks profit. Subsidized by the government, it wouldn't be the first capitalist enterprise to rely heavily on government funding.
Amtrak is a federally chartered corporation, operated and managed as a for-profit company, but with the U.S. government as its controlling shareholder. The Amtrak Board of Directors are appointed by the president of the United States and confirmed by the U.S. Senate.
https://www.amtrak.com/stakeholder-faqs
I'd consider it state capitalism. A private for-profit company owns the means of production/capital but that company is government owned.
That being said though, I don't think its fair to blame Amtrak being a private company as the reason Amtrak has high prices. Amtrak is at the mercy of freight rail as Amtrak only owns 3% of the rails they operate on. They pretty much have to pay whatever the freight companies say and freight is given preference over passenger trains. Amtrak has never made a profit since its creation.
State capitalism is an oxymoron. Socialism(not Marxism which is more specific) is the collective control of the means of production, collective control includes state control, I(and I think most people) consider capitalism and socialism to be antonyms, so I don't think you can really say state control of something is anywhere near capitalism.
Amtrak is at the mercy of freight rail as Amtrak only owns 3% of the rails they operate on.
That is true, but I don't think anything stopped them from building their own track like Brightline is doing.