Does everyone that goes to the same bank have the same routing number?
No, the routing number is not defined by the bank but by the state and the type of institution where the account was opened. “Does everyone have the same routing number at the same bank” sounds impossible everything would be bewildering. You may even have different routing numbers in the same bank: for checks and deposits separately.
Answer the question what is the safest way to dispose of old bank account statements?