>Don't they presumably have a US alien ID (Green Card or otherwise) and/or a foreign passport

There are millions of visa-holders who live in the US but don't have green cards. Carrying around a foreign passport for everyday identification is an absolutely terrible idea because of the risk of theft or loss. Losing a visa-containing foreign passport can necessitate getting an emergency travel document from your foreign-country consulate, traveling back to your home country, getting a new passport, getting an appointment at the US embassy in your home country, getting a new visa issued in your new passport, then finally traveling back to the US. The whole process could take weeks or months and cost thousands of dollars.

My wife is foreign-born, but was in the US as a non-resident when our hearts turned proximity into much more. Since she was already here the whole application process took place in the US. It was just under two years from when we married until she had a green card. (And note that there is also a delay from being granted a green card to actually having one in one's possession.)

Foreign passport and non-driver's ID were her only options--and passports are a nuisance to carry around and a PITA if lost/stolen.

(And, yes, I think that just under two years bit was not by chance. They were definitely looking for signs of fraud and I'm sure they finally approved it just before the point where they would have had to issue a permanent one rather than a conditional one.)