All banks and bank-like institutions you might have access to will comply with the same anti money-laundering and know-your-customer regulations. It's not really up to the individual bank to decide.
The "Swiss bank account" is largely a thing of the past, as they are forced into data-sharing agreements with the United States and possibly others. And even so, you need to be a Swiss national to qualify for a bank account there.
You can use cash as much as possible when it's an option, of course. It's less convenient, but privacy often is.
Due to Apple's policies, there are no good ones. The least bad one is called Onion Browser and is recommended by the Tor Project.