What "offshore" actually means for card processing
Card schemes license acquirers by region. A UK acquirer serves UK merchants, an EEA acquirer serves EEA merchants under passporting rules, and acquirers in places such as Mauritius, Cyprus, Malta or the Caribbean may serve companies registered elsewhere. When the acquirer's licence region differs from the merchant's customer base, the account is usually described as offshore or cross-border.
The practical effect is on pricing and controls, not legality. Cross-border transactions can carry higher interchange and scheme fees, and the acquirer needs to be permitted by the schemes to board merchants from your company's country. An acquirer that is not permitted to do so cannot fix that with a higher fee.