Oddly, I can't remember anyone ever asking me for my ID on the train - just the ticket. And if you purchase the ticket online, you can print it out either at home or at a kiosk with just the credit card or confirmation number.
> what they were asking was illegal.
Are you sure it is? They're a private company - on what basis is it illegal?
They are structured as a for-profit corporation, but they are a creation of a specific federal law, funded by the federal government as its sole active investor, and created as a federalization of the failing passenger rail industry.
They are very much an organ of the government.
I was once asked for my AAA card when using that discount on the regional, but even that is extremely rare.
Once in the South, I was fine just with an online receipt. Another time, going from LA to SF, I had to buy another ticket because my name was misspelled(seriously).
I think maybe the tickets said something about "you may need to show ID to the ticket guy on the train", but I never had that happen.