Amtrak is owned by the government but is not really part of it in the sense that the Department of Transportation is. It is only partially funded by the federal government. A vast majority of it is really funded by its users and state governments. The chartering instrument of Amtrak explicitly says that it should be run as a for profit business, which contrary to the best wishes and fantasies of some railfans, sounds way more like a private business than a government department.
"For profit" can mean a lot of things. For example, it could simply mean that the Amtrak's revenues should exceed its expenses only to the point that no further taxpayer subsidy is needed, which is I think is what Congress was thinking when they created the company in the first place. Or it could mean that it should be managed to infinitely maximize its profit and value to the shareholders, as most private corporations do nowadays.
Because (as far as I know) Amtrak is owned 100% by the U.S. government, and its not traded on any stock market anyway, "maximizing its value" seems to be sort of silly, unless the government want to sell it for a short term cash windfall. (But every instance I've heard about when the government sells a public asset, they usually sell it cheap to a politically well-connected buyer, and the government doesn't make the financial windfall it should get.) "Maximizing profit" might be more reasonable, because the company could use that to expand service. Of course, most private companies are managed today to spend as little as possible on making the product or service they sell, and the maximized profit, if it is realized, is disproportionately distributed to the top executives and shareholders, and many times, the shareholders get screwed, too.
I think that the authorizing legislation and amended charter of Amtrak should be more explicit about the reason for Amtrak's existence, and why it needs to be a government company rather than a purely private company. I would think this would take the form of a finding that a passenger rail system is a national need for reasons of:
Removing cars from the highways (reducing both traffic congestion and petroleum consumption, and greenhouse gas emissions);
Enabling denser, walkable development of towns and cities (reducing auto traffic and thus traffic congestion, petroleum consumption and greenhouse gas emissions);
Providing alternative mobility for people in rural towns, especially for people who can't or won't drive their own cars;
Providing alternative mobility for the significant minority of the population who can't or won't drive for medical or other reasons;
etc.
The mission of the company should thus be to maximize ridership and areal coverage of the service and increase revenue, yet keep the cost related to the revenue low enough that government subsidies aren't needed. That's what "profitable" should mean.