First of all: most of europe was part of the roman empire. Even parts that weren't were influenced by it. That lead to the next connection: the christian chruch was distributed by the roman empire and won might and stayed after the roman empire fell. Even for remote parts that still were not christianized the christian conquerors did that more than thousand years ago. So until the protestant reformation most parts of europe were theological under the pope. Only in eastern europe you had the orthodox church. And last but not least: in the middle ages all the aristocracy made weddings with each other. Queen Mary in Britain had an spanish husband. Catherine the Great who lead Russia for a time came from germany. And so on.

So history of 2000 years connects europe. That isn't the case for other continents. You say you're from south america? Well, the connections from 2000 years ago were destroyed at least after the european colonization.

In Sout America, and in Latin America in general, there`s a strong sense of "us" when talking about other countries. And the pre-colonization civilizations are to this day very celebrated as a form of resistance to the european colonizator. With the big excepetion of Brazil, South America has a strong sense of continental identity. It can be attributed for them to have been colonized by Spain, sharing the same colonizator and the same language helped unify this feeling. Even the independence movements had a common hero in Simon Bolivar figure.

Brazil on other hand is a completely different thing, we were colonized by Portugal, speak a very different language and along our history some regions were under heavy influence of France and Holland. Heck, even inside our own territory cultural differences are huge, if we put an average Salvador citizen, an Porto Alegre citizen and Sao Paulo citizen their culture and ethincity couldd be as different that some could even think they are from completely different countries, Brazil is far from having an unified national identity aside some trivialities. And we are so not interested in the rest of latin america that it is much easier to find a brazillian that speaks english than spanish.

I don't think there really is a unified european feeling. There's a deep divide between western and eastern Europe and a less drastic, but noticalble divide between nothern and southern Europe.
The nordic coutries as well as the north of Germany and the north of Britain share a lot of cultual ancestry being dominted by the vikings and keeping (knowingly or not) a lot of the old traditions alive to this date.

Much of the reason we celbrate christmas in winter instead of summer, where the earliest sources actually put the birth of christ, is because in order to break down Viking resistance, the romans assimilated huge parts of their culture and merged it with christianity. Sort of like: 'OH! Your most important holyday is the winter solistice and bringing light into the world on it's darkest day? Boy, do I have a story for you! By pure coincedence our lord and savior just happens to be born on the exact same day, and there was this star and everything!'
We actually see very similar things in Hatian and New Orleans christianity, where it got combined with Voodoo.

Then like other people have mentioned the roman invsion and christianisation of europe did a lot to forge a similar cultural heritage over a lot of Europe.

Nevertheless many of the country-to-country diffrences are staggering.

So you agree that a strong part of the current cultural identity in south america is formed by the european invasion. That is not as long as 2000 years, because europe discovered America a lot later.

Though people will probably feel more connected to other European countries than to countries elsewhere, a real "us" feeling doesn't really, mildly at best, exists continent-wide. There are pockets of countries that feel more connected.

Speaking from a Dutch perspective there's obviously the BeNeLux, for me being in either Belgium or Luxembourg is almost like being in my own country (well, they do have prettier nature). In a broader sense you could say we stick mostly to the other North-Western European countries because our cultures and values are mostly the same. Countries do usually cling the most to their direct neighbours though and the longer the distance, the smaller the connection. On a deeper level it's probably because we all share thousands of years of history and mostly common origins. Countries (Kingdoms, Empires, revolutions) rose and fell, we owned each others lands, everyone's royalty is basically one big family and at one time two countries would depend on each other while at another they were at odds. All our histories are intertwined.

Still, like I said, we are all Dutch, Belgian, English, German, Swedish, Polish, Russian, French, Spanish, Italian, Greek and all the others first and foremost, and European second. Differences between countries are big, even neighbouring ones. I could probably show you any stereotypical photo of any random country and you'd know which one it is.

Id say its our values & culture.

Equal rights, freedom's of various nature, ect.

People don't feel European, they feel like they belong to their respective sovereign nations or sometimes, rarely, to some ethnic tribes within states. End of story.

Divide between west and east have more to do with economic differencies, I think. Of course the ethnic differencies between people with eastern Europe having lots of slavic population and western not having. Which takes us to ethnic differencies between northern and southern Europe, when looking at the northern Europe pespective, Spanish/Portugese and Greece/Italian people aren't even white.