Bank secrecy is a legal and sometimes constitutional option in some jurisdictions that apply a certain privacy "veil" to bank account information. These privacy options vary from jurisdiction to jurisdiction, and not all are the same, nor are they suitable for all types of accounts. Indeed, a private individual opening an account in a country that enjoys bank secrecy will only be able to benefit from it if they also move there with tax residency. Tax residency is the legal basis on which offshore banks determine whether an individual can benefit from bank secrecy or must instead exchange banking information with their country of origin.
Only those who genuinely and legally move to another country will be able to benefit from that country's bank secrecy. Otherwise, banks will be required to exchange information with European tax authorities, following the same process that already occurs between institutions and tax authorities within the European Union.