An offshore account is an account you have in a country where you do not live. If you live in France and open an account with a Cyprus bank, that account is offshore to you. For someone living in Cyprus, the same account is a local account.
In Cyprus, that could be a current account, foreign-currency account, payment account, or savings account. “Offshore” describes where the account is held in relation to you, not a special type of banking product.
You will still need to complete the bank’s normal identity, tax-residency, and compliance checks. Cyprus also participates in the Common Reporting Standard, so account information may be reported to the relevant tax authorities where the rules require it.