If you’re in the US, your bank knows way more about you than that and it’s naive to believe otherwise. A lack of credit doesn’t mean a lack of tracking; it just means your data is being pulled from elsewhere.
If you’re not in the US, you might have a better chance at privacy.
Do you have a drivers license? A social security number? A phone number that you’ve used for anything else? Utility bills? Relatives? A car? Other large property?
Cash doesn’t mean shit unless you pay for everything in cash and never use the same info (including name, address, phone number, social, etc) for everything.
No car, no property, no phone number.
I don’t think a bank having my social security number is a privacy risk.
I do not give my name when I buy groceries or 99% of purchases. I don’t see your point.