Many people think that a secret account is some kind of hidden option. In reality, it's just marketing. I read about
secret bank account here and came across some real stories. For example, even if an account is opened in a jurisdiction with increased confidentiality, the bank is still required to know its customer and will request documents. The most you can achieve is that the data will not be automatically transferred to other countries without good reason. More interesting examples are given in the article, which I will leave for you to read.