Eu banks typically use a MasterCard or Visa partnership for their credit cards. The EU bank might issue the credit card to their customer, but the actual payment processor is an american company. If MasterCard/visa starts blocking certain payments, then there's nothing that the EU bank can do about it.

You can know which payment processor your bank's credit cards uses by the presence of a small logo on the front of the card. 2 overlapping red and orange circles = mastercard network.

As for car rental companies, Hertz has some wonderfully twisted logic on their Belgian site where they say that they accept debet payments from any eu bank card, as long as the card has the visa or Mastercard logo. In other words, they only accept Mastercard or Visa payments, not eu debet payments.

I mean if you don't understand things you should ask, not assume others are in the wrong. Yeah it would be pretty stupid to be forced to have US accounts to rent cars in Europe, luckily we're not THAT stupid.

VISA and MasterCard are practically a monopoly on credit card circuits, your parents' cards may be issued by a local bank but there's a 95% chance they've got one of these two companies' logo printed on them, and out of every payment they make, these AMERICAN companies get 2% (blah blah cashback, blah blah terms and conditions), because they are made on their circuits!
You may also have debit cards that DON'T have those logos, but debit cards can't be used for car rentals.