At Desjardin I showed my dutch passport and driver's license. No problems, no SIN card needed. The SIN card is not even an official ID.
The only thing is that they refuse to give me a Visa credit card, until I have my Québec Selection Certificate or Permanent Residency. No problem: still got my european MasterCard and I can (limitedly) use my Desjardin debit card.
The only thing is that banking in Canada is pretty expensive and their online banking site is anything but easy to find your way in. On the other hand: my bank is a co-op, so my money doesn't go to my director's seventh Bentley/Porsche/Lexus/other overpriced car.