well when I moved to the UK I had a very hard time to get a bank account.
I needed a national insurance number but because I had just arrived in The UK, I first needed a job to get a national insurance number. When I started work my employer rang up and got me a temporary one so that I could be put on the payroll, and made an appointment for me to have an interview to get a proper one.
At the interview I needed proof of work AND proof of address. So I then had to register for council tax. Which meant another interview process where I had to show my passport and flat rental agreement.
It took about 6 weeks to get my national insurance number sorted and get registered for council tax.
Once I had all that sorted I then went to the bank. I needed my passport, to show that I was an European citizen and could legally work and live in the UK, I needed my national insurance number, I needed my council tax documents to show proof of address and I needed payslips to show that I had a job.
Im guessing its the same type of process and similar types of documents are needed to open a bank account in America.