Not in the USA.... not until they're here with bonafide, valid, undisputed papers.
Not that I approve of illegal aliens, nor this bank and its sleazy marketing gimmick, but it is true that we Americans can open bank accounts in just about any country that allows it.
Heck, I've had bank accounts in Singapore and Vietnam, of all places.
And there are 100 percent legitimate international banks (e.g., Citibank, HSBC) that will let Americans open accounts in more than 100 countries. You (we, I) can even open a Channel Islands account online, over the Internet (subject to providing followup ID, bank refs, address confirmation, etc).
The illegal alien issue is separate from the notion of banking practices, although there clearly is something wrong with a bank accepting these bogus Matricula IDs.