You can obtain a mortgage from the bank without providing your SIN. The bank can obtain your credit bureau file without your SIN.
You *are* required to provide your SIN to the bank though if you open accounts that generate interest or income. This is for tax purposes.
For more information, see this page on the Government of Canada Website
Specifically:
When you don’t have to provide your SIN
Some private-sector organizations may ask for your SIN. This practice is strongly discouraged, but it is not illegal.
Here are examples of when you don’t have to give your SIN:
proving your identity (except for specific government programs)
completing a job application before you get the job
completing an application to rent a property
negotiating a lease with a landlord
completing a credit card application
cashing a cheque
completing some banking transactions (mortgage, line of credit, loan)
completing a medical questionnaire
renting a car
subscribing to long-distance or cellular telephone services
writing a will
applying to a university or college