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.