I read in Business Week a while ago that undocumented hispanic people are a target market for the mortgage industry. And the matricula card that you mentioned was cited as an acceptable ID to purchase mortgages.

So I suppose policies vary by bank, by region and by the type of financial service.

Having once been a Chinese immigrant (I think I'm pretty well assimilated by now), I am surprised that you friend even accepted a check, went to the bank and presented her various forms of ID. It used to be that new Chinese immigrants, documented or not, would never engage with the establishment, remembering what woe can befall them for invoking the establishment 'back home.'