I think that a lot of people equate the euro to the European Union, and the EU to peace and prosperity in western Europe... Some people also conflate the EU with the Council of Europe while these are two different institutions (whithout any link in membership).

To a certain degree, this is true. But it has always seemed to me to be forgetful of others stability factors that maintain peace between the western Europe nations: nuclear weapons in France and Britain for a start.