Do people from India speak Indian? Do Chinese people speak Chinese?
As said above, The whole nation state thing is a modern construct. A lot of the problems in Africa are the result of Europeans drawing up borders with little regard for local tribal lands.
It may be worth noting that Europe has, basically, settled on territorial boundaries that more or less match linguistic boundaries. Austria is the last remaining holdout. But the breakup of Yugoslavia has pushed back in the opposite direction. Before 1990, people in ex-Yugoslavia pretty much spoke a mutually intelligible language. Now, Slovenians and Bosnians and Croatians and Serbians are different countries speaking a common language. It’s called Serbo-Croatian, distinguished largely by the alphabet of choice, similar to Urdu-Hindi of the Indian subcontinent.
But historically, there has never been a “rule” that national borders need to coincide with linguistic borders. Kings can change the boundaries of their kingdoms, but changing the languages of their subjects is a slow and grinding process that seldom materializes.
I do wonder whether “Germany” (as a spatial and cultural entity) is as diverse and in-fighty as “Italy” is. In my mind, Germany actually managed to become a unified country over the last two centuries whereas Italy has only learned how to pretend to be one over the same timespan ; but I have no idea how close to reality that is.
Apart WASPs, perhaps 
The French enjoy calling everyone in the anglosphere ‘anglo-saxons’, usually when they are disparaging some element of our collective culture.
Spain, France, Andorra, Switzerland, Liechtenstein, Italy…
Germany and Italy, as modern sovereign entities, both unified in the middle of the 19th century. By that time, the standardization of the Italian and German languages had already been under way for several centuries.
Belgium, Ireland, Moldova…
Slightly off topic, but it was thanks to a reader’s message to Cecil that was included in a Straight Dope column (scroll to bottom) that I realized that Charlemagne – that is, Karl der Gröss – was more “German” than “French”, and I noticed many modern French words derive from Frankish (a Germanic dialect).
Well, Liechtenstein is actually the only historically 100 percent German-speaking country in the world.
The amount of variation in what we might call the well-known European languages (English, French, German, Italian, Dutch, and Spanish), in all the places where they are spoken both near their place of origin and in the places far from those places, is greater than you might expect. It would take a while to explain just how much variation there is. You might look at the relevant webpages in the websites Ethnologue or Glottolog for a detailed explanation of that variation. In some cases it’s usually thought of as being dialects that are at the edge of mutual intelligibility, while in other cases it’s clearly so different that it’s considered different languages:
The German word for France is ‘Frankreich’ (Frankish Empire) 
By the way, the most homogeneous country in Europe - perhaps in the world - ethnically and linguistically (i. e. one country, one ethnicity, one language) is the Czech Republic.
Just to add to the discussion, South Tyrol is a province in northern Italy where the majority of people speak German (it’s Italian name is Alto Adige). It is part of a historical entity (County of Tyrol) that stretched into present-day Austria, including the Brenner Pass and Innsbruck. I visited the area in the early 90s and road signs are in Italian and German. Another example of borders not reflecting local language.
Well, now. Ever hear of Sudetenland?
Yes, all the Germans who used to live there (like my neighbour) were kicked out (not just from the Sudetenland, mind you, but also from the German speaking parts in other parts of the country like Moravia). That’s the very reason why the Czech Republic is so ethnically homogeneous today.
The main point is that there was a large collection of various kingdoms and principalities with varying degrees of (nominal) independence and interdependence - they were only fully united into “Germany” in the later 1800’s. (Note that “mad King Ludwig” of fancy castle fame was still “King of Bavaria” until almost 1900 and the Bavarian monarchy crumbled only at the end of WWII, although technically it was a province of the German Empire before that. )
Also there is a regional variation in language - as others point out, different words for some things (Think “teevee” vs. “telly” or “boot” vs “trunk”.) I believe someone on one thread once mentioned that “country bumpkin” accent in movies tended to be Bavarian, much as southern USA accent tends to be. I assume the accents are regional enough it would be analogous to England, where the accent defines what region you grew up in, in a very small country?
Central control, central source of modern media and centralized “authority” for the language probably have done a lot, like other areas, to help standardize a very regional diversity. I understand it’s the same in Italy, where a diverse collection of small states were united in the late 1800’s, but regional accents of Italian are quite pronounced. Even Our French teacher mentioned that France has a number of strong regional accents, despite being a kingdom under central control for almost a millennia.
There is an interesting Wikipedia article about the “Republic of German-Austria” (the official name of the country from 1918 until 1919):
If you wanna tour Berlin, You’d better get High
If you wanna get up and take a plane, you’d better get High
The country seems so cool, and I know why
They all get High
They all get High
They all get High
Thinking 'bout Oktoberfest? You’d better get High
The beer is so great and Dutch is no good - Better get High!
We are taking it next semester and I know why
Because they get High
Because they get High
Because they get High
Lots of Germans go to work because they are High
Hans just got a new promotion because he is High
Now he’s building cars and I know why
Because he got High
Because he got High
Because he got High
I was gonna bail out Greece but they won’t get High
Their country is in such a mess because they’re not High
Lebensraum for us and I know why
Because we are High
Because we are High
Because we are High
And it’s not in Germany, I think is the point. Territorial borders do not reflect linguistic.
I looked up the Czech Republic on Wikipedia. According to the numbers there, 64% of Czechs consider themselves ethnically Czech.
By way of comparison, North Korea is comprised pretty much exclusively of Koreans, speaking Korean. There is a very small number of Chinese and Japanese, but I’d imagine the Czech Republic’s ethnic variety probably beats North Korea’s fifty to one.