Since there haven't been qualified answers yet and I'm not sure mine would fare better, I'll link to questions going in the same direction.
Additionally, while not answering your actual question, there arguably had been a unified German State in 1848/49, albeit only for a year. The German Confederation briefly became the German Empire (not to be confused with empire of 1871), an implementation of the "Kleindeutsche Lösung", a German nation state excluding Austria.
Edit: for the case that people might only read my answer, I'd recommend skipping onto u/totally__paranoid below whose answer is far more indepth dealing with immigrant Germans in 1800's.
So I think answering for the word "German" is quite difficult at it is the English language word for the nationality, but "Deutsche" is rather more readily available. Whether one can directly correlate "German" and "Deutsche" in the 1800's is a question I don't really have an answer for.
Certainly by the 1500's there was an identity of "Deutsche" that was prevalent at least amongst the aristocracy / wealthy classes, one of Martin Luther's pamplets was titled An den christlichen Adel deutscher Nation.
More specifically, around 43 years prior to 1858, after the end of the Napoleonic wars, the various nations that would go on to make up Germany, Luxembourg, Austria, Czech Republic and Slovenia (and large swathes of Poland) were confederated in the "Deutsche Bund", so certainly by 1858 the idea of "being Deutsche" was certainly around and much of the political discussions in the 1840's was over whether Germany would form "kleindeutsche" or "grossdeutche" (basically would the German nation form the smaller German country under Prussian rule, or the larger German country under Austrian rule).
The confederation fell apart in 1866 with war between Prussia and Austria. At the end of that war, what is now North Germany became a federal nation, with the southern states largely staying independent until 1871. But even then, when Ludwig II of Bavaria approached King Wilhelm for Bavaria to join this federal state, it was with the mind of forming a unified German Empire, from the Kaiserbrief where Ludwig II and Otto von Bismark urged King Wilhelm to assume the title of German emperor "Nach dem Beitritte Süddeutschlands zum deutschen Verfassungsbündnis werden Ew. Majestät übertragenen Präsidialrechte über alle deutschen Staaten sich erstrecken. "; "With the acceptance of southern Germany into the German Constitutional alliance, Your Magesty will extend presidential rights across all german states".
So therefore the idea of being "German" [Deutsche] was certainly around, was certainly wide spread amongst the aristocracy and powerful families of the 1800's, and probably before even the 1500's.
In Django Unchained, to quote from the specific wiki on the character's background "King Schultz came from a wealthy family in Germany (presumably Düsseldorf, Rhine Province, Kingdom of Prussia) and is highly cultured as well as very intelligent". So I would think that it is very reasonable for someone of that background to refer to themselves as German.
Would the "average man on the street" do so? That is a very different and very interesting question as well, that sort of nationalism of the common man was something that played an undercurrent in many European Kingdoms from around the 1300's, but ideas of nations were not as concrete as they are now. At the start of the 1800s, an average person from Düsseldorf would probably be as likely to identify as from that City as from any of the proto-german dukedoms. The idea of the "everyman" thinking of themselves as from that nation, at least in Germany, started to arise during this actual period of the mid 1850's, with Bismark's drive to rapidly industrialise first Prussia, then Northern Germany, then the German Empire, thus the "everyman" born at the same time as Schultz in Dusseldorf (which, one should note, only became part of the Kingdom of Prussia in 1815), who then emigrated to America, would potentially be far more unlikely to consider themselves German [Deutsche] (though the idea would not be completely foreign to them).
Hi, great question! I'm interpreting your question as asking about German-speaking immigrants in the US, but let me know if you'd like me to expand beyond that.
The short answer is yes, a German-speaking person in the United States in this period would absolutely refer to themselves as a German. The long answer is that "German" would be only one of many identities, and many factors would go into how they ranked their "German" identity among all of their others. Those other identities - specific country of origin, religion, and associational/political ties, to name a few - were equally important in German-American life, and individual German-Americans would prioritize different aspects. The context matters tremendously, too. In the conversation you're referencing, Schultz is speaking to a non-German, so identifying as a German makes complete sense. If Schultz were introducing himself to another German immigrant, it's entirely possible that he would introduce himself as a German, though it would be likely for him to emphasize some other element of his identity.
In fact, this ambiguity in terms of immigrants' most important self-identifiers is exactly what makes the study of German-Americans so complicated. In Bonds of Loyalty, Frederick Luebke, something of a grandfather of this field, broadly breaks down German-speaking immigrants as "soul Germans" and "stomach Germans," depending on where they most vividly displayed a recognizable sense of "Germanness," and these cultural displays were the focus of a lot of the ethnic studies work of the 1980s-1990s (see, for instance, Kathleen Neils Conzen's work).
So how did scholars decide that this vast array of people count as German? First, because they were understood to be "German" by their 19th century contemporaries. Anglophone Americans famously referred to German immigrants as "Dutch," but this was a casual (if not pejorative) nickname, and in polite conversation and official documents the term was "German." This was widespread and normal. For example, as you probably know, somewhere around 3/4 of 19th century immigrants entered through New York, where entry was governed by the Commissioners of Emigration of the State of New York. In their annual breakdowns of entrants by country of origin that they submitted to the state legislature, they actually listed all German-speaking immigrants as from "Germany," even before 1871! (This seems to include Austrians too, although not Swiss Germans).
More importantly, though, German immigrants called themselves that. Stanley Nadel notes in his book on German-speaking New York City (Little Germany: Ethnicity, Religion, and Class in New York City, 1845-1880) that settlement patterns and associational life certainly were primarily determined by kinship, regional, religious, and dialectic background, but all of this coalesced in a "German" New York. The label for the largest German-speaking neighborhood, Little Germany, was also used in its literal German translation (Kleindeutschland), This self-adoption of German as a catch all is common in the volumes of personal records, newspapers, and popular fiction generated in this period. The usage of Deutschland to refer to the German-speaking lands and Deutschen to refer to German-speakers was so widespread and so casual in this period that it's frankly hard to know what to cite, so I'll give you examples of its usage in different contexts. All of these excerpts from the Deutsche Auswandererbriefesammlung (DABS) at the Forschungsbibliothek Gotha of the University of Erfurt:
On the flip side, emphases on on sub-German identities generally only appear in these letters when they're writing about kinship or community networks.
Sorry for the wall of text, this was a long way of saying that yes, it's accurate for Schultz to have described himself as a German.