There are 1 or 2 minor exceptions to the flight plan requirement, but mainly restricted to short hops between flying clubs close to a border.
I've flown between small strips in France, Italy, Germany and Sweden (Schengen but not EU) with no formality except activating your flight plan when airborne and closing it just before landing; all perfectly legal.
If only flight plans/ customs / immigration / special branch could share information and have a single interface, wouldn't life be a whole lot easier?