FDIC Regulation 500 prohibits discrimination in making loans on the basis of "National origin" but not on the basis of immigration status. This story from The Nation says that Bank of America is denying accounts to non-citizens, and arguing that it is legal because of increased risks, although there are current court challenges to this. Perez v. Wells Fargo Bank, N.A is a case now pending challenging loan denials based on immigration status. This has particularly come up in regard to DACA recipients, rather than people with LPR status.
The US "public accommodation" laws probably do not apply, as a bank is not usually considered a place of public accommodation. Any specific state laws prohibiting discrimination on the basis of immigration status might apply.
In short, this is an issue still not clearly settled. There seems to be no law or regulation requiring banks to ask for citizenship information, much less to deny accounts based on it, and it would be well to seek a bank with a different policy if possible.
The above is very US-specific. Many countries do limit banking access based on citizenship, i understand.
I am not a lawyer, and this is not legal advice. Before challenging any bank action, you may well wish to seek advice from a lawyer.