How you open an account depends upon the jurisdiction where the account is established. Some countries regulate their bank accounts rather strictly because they are identified as potential narco-states by the United States government, so they, in conjunction with some of their financial institutions and their government, construct rules and regulations. Some of them include producing identification, providing copies of reference letters from known legal sources, etc.
Others won't allow a foreigner to actually open a corporation. What you end up doing is going through law firms that have established the corporations with local boards of directors and then you become what's known as the beneficial owner of that corporation by buying into a trust at the various law firms, so if you need to do it without ID, it can be done. If you need to do it with ID, it can be done. You just have to pick the country in which you want to do it.