>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.
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.