Let's start from the basics: All nation states are invented. You've mentioned Italy and Germany. Let me add the UK (England conquered others), Spain (Castille and the Kingdom of Aragon merged, then Castille imposed their culture over the years), France (many different tribes and many different languages apart from French), Belgium (what is Belgium?), etc
Even some more homogeneous (apparently) countries are melting pots. Romania: Some regions speak only Hungarian (and they're not bordering Hungary) and some others speak also Ukrainian. Their borders have changed a lot in the last 150 years. Hungary itself is nothing like it used to be (and it was the same country / empire with Austria... go figure!). Let's also not start with disastrous examples like the former Yugoslavia...
The difference is that the US assumed and took advantage of the "melting pot" idea whereas nationalistic views in Europe have been making the world think we have some special character or culture (and each of us different than its neighbour). Two World Wars mostly based on nationalism haven't helped either.
In many European countries it is frowned upon to worship the national flag or at least it's considered potentially close to extreme right wing and racism. Why? Because as opposed to the US flag which represents a melting pot and freedom, European flags represent nationalistic ideas. Ask and Englishman about Saint George's cross or a Spaniard about their flag. In both cases they'll only wave the flag during sporting events like the current Eurocup. Outside that, the above applies.
> Though I do think it's telling that a good number of things you offered as components of a shared European identity were invented in America.
Globalisation. Also, America didn't just happen. America is a branch of older European cultures (British, Irish, Dutch, Spanish and French at least), so in a sense the American culture is child of European culture, hence why it's not difficult for us Europeans to make it our own too.
American ideas and ideals come from Europe and / or Christianity. Some of them even predate the idea of Europe, coming from the Romans or Greeks. This is a long heritage than you guys and us have shared for long time. Globalisation is just making the rest.
> Do actual Europeans say that? Sounds more like the typical American who just got back from a vacation in Europe.
Not unheard of, specially if we're talking to someone not European. A clear example is political views: In Europe we consider American elections a choice between right wing and righter wing ;) whereas some of our social democratic (or labour) parties might look communist to you guys.
Times change and this might not be true anymore in a few decades, but a majority of Europeans would always choose Democrats over Republicans (and Bernie over Hillary).