Even people very convinced of the EU don't really think of themselves as European first. Try to ask them where they come from when the EU is not a topic and you get the real answer.
There is no overreaching European media that could form stronger bonds. Cultures and languages are too diverse and that is probably the main difference to the US.
I think there is thinking of European as continent level. That is Europeans are not say Asian or African, or neither Americans. So at global scale, we are one group distinct from other continentals groups. If those continents even have any shared belonging in a group. But we certainly are not them.
There is no overreaching European media that could form stronger bonds. Cultures and languages are too diverse and that is probably the main difference to the US.