It’s likely about control and ownership

There was once a VISA Europe branch that was separate from VISA .inc, it’s not the case anymore since 2015 when VISA bought back ViSA Europe.

There are countries where debit cards are also compatible with a local system and in those countries, banks make the POS devices use that local system by default (for cost cuttings) if the card is compatible and only use Visa/MasterCard if the card is not compatible.

There’s normally two different fees when doing a card payment: the interchange fee (Visa, Mastercard, and the couple others), and the processing fee (Square, Stripe, PayPal, whatever).

Nowadays, interchange fees are capped to like 0.3% in the EU, but this wasn’t always the case. And for non-EU, the interchange fees are usually much higher.

So even if both a store and their payment processor is EU-based, Visa/MC is still giving at least 0.3% of your payment sum in most cases.