The currency of much of Europe is the euro. It is not the euro dollar.
A euro dollar is a deposit of U.S currency in a bank outside the U.S--usually to get a higher return margin.
If you walk into a bank in Europe and say you want to buy eurodollars, they might think you want to buy a futures contract on eurodollars and not the currency you need to buy the cuckoo clock you saw in the store window.
Just because we call our currency dollars, doesn't mean the Europeans do too. Afterall, should Mexicans call it a "euro peso," or the Japanese a "euro yen."