If you don't qualify for an SSN you can get an ITIN (Individual Taxpayer Identification Number) that can be used for some things in place of an SSN, but it's not a "first class" identifier. I think an ITIN is fine for setting up a bank account, but it's been a while since I did it.

I'm going through the bank opening process as we speak with SVB however I'm from the UK and with a new branch of SVB being opened in London, I feel the OP will be in a different position as he isn't from the UK.

Yes, you don't need an ITIN either. A passport and another (any country) credit card with your name on it will be enough.

People on a US student visa often aren't required to file US taxes and therefore do not need an ITIN.