I think we have to distinguish between a general cultural European identity (which is actually not that new), and an artificially created EU identity. I guess nobody in the world identifies himself with the EU. Or would someone from the EU, when abroad, seriously introduce himself with coming from the EU if asked where he's from???
Comparisons between a general European culture and anything else (LatAm, African, Arabic, Indian, in some cases even Russian or Pan-Anglican) on the other hand is more likely.
The first time I felt a real European identity was when I was in East Africa, living among Africans, Indians and Arabs (and Cuban collegues). After intensive studies of pre- and post-colonial politics, history, religion, habits and (pop-)culture I started to ask myself what my roots are and what my own identity is. Sometimes I was getting really desperate in understanding the other people's way of thinking, whether it was economics, hierarchy, moral or any other sort of irrational thinking out of my perspective. One day, after I haven't seen a European for over a week, I walked down a street where I went into a barber shop and surprisingly met an Italian inside. Even though I never felt Germans and Italians as that alike before, all of a sudden now I felt understood by someone, followed by the thought "I am a European!".
Additionally, in Africa, European culture was often over-generalized. There is a term "Mzungu" (literally "aimless wanderer", but in the sense of "European"), applying for all Europeans AND those who follow the European culture and habits. Which sometimes also lead to the strange situation that even Afro-Americans who have lost their African cultural roots were considered as "Wazungu" (plural). Anecdote: an African friend gave me a CD for my birthday with alleged European Music on it, and I was surprised to find Snoop Dogg and 50cent as I listened to it... Strangely, American Hip Hop was regarded as a product of former African slaves exclusively influenced by European cultural heritage.
To the rest of the world Europeans often present a very paradox picture:
-once they tried to impose Christianity on everyone, now all of a sudden they say God doesn't exist
-once they fostered the sanctimonious little family with two parents and one to three kids, now they say we have to abandon families and live in sexual liberty
-once they talked about industry and progress, now they talk strange things about ecological sustainability
-their monarchs and dictators showed how to be successful and rule the world, now they talk about freedom and democracy and that stuff
To an outsider it itself seems irrational and inconsistend! But it shows very well that also European identity is not static, it changes over the centuries and even decades. So besides it's roots in Christianity and the Greco-Roman world, also these rapid shifts in culture and mentality is what makes European culture European.