Well, are the Balkans and environs and Ukraine part of your Europe or not? Because for those regions, the Ottoman Empire (and also the Golden Horde/Crimean Tatars more to the East) is a rather defining part of history. In various ways. (This old diary of mine might show you history you wouldn't expect.)

So what does our commitment to the European project say about us:  It says we want to put an end to the wars between us, to transcend the national rivalries that have in part caused those wars, we want to work more closely together for our mutual benefit and prosperity, we want to help those regions and sectors of society which have lagged behind, we want to put behind us those parts of our history - religious intolerance, racism, class division and discrimination of all kinds which have been used to dehumanise others, and we want to project those values more effectively onto the world stage.

Is that not what our European Identity is about?

Precisely. Or, to put it another way, European identity is in an entanglement of inter-relations, not in a common trait.

_{*Lunatic*, n.
One whose delusions are out of fashion.}