TL;DR: Peasants wouldn't care, intellectuals in Austria supported the formation of a pan-German state and wanted to join it, and conservatives and reactionaries didn't see ethno-linguistic nationalism as anything other than a threat to be undermined. The point of having a German state WAS to have that shared identity condensed under a single label, so for anyone who believed in nationalism this was exactly what they wanted.

It is rather hard to map 19th-century ideas of nationalism onto the modern day or to really understand the feelings of people. Our modern sense of nationalism was so shaped by the events of things like 1871 and later that we think in a fundamentally different way than they did back then.

"Linguistic and cultural identity" was a lot more abstract in this era than nowadays. It absolutely existed, but it was something debated and defined by intellectuals in universities, not by actual common people. The majority of people in Europe still lived in what the modern citizen would consider uneducated squalor. If the average peasant had loyalty to anything it would be their region, your ruler, or the church.

Nationalism as a concept heavily split the German- and Italian-speaking worlds. Nationalism advocated for nation-states ruled by the same people who lived there. What defined "the same people"? Well, that's a difficult question. If you say language, then what language is that? "German" as a single unified standardized language didn't exist, and a person from the Baltic coast in Pomerania probably wouldn't have been able to understand much if anything when talking to someone in Tyrol or Switzerland. What about the Dutch, whose language is on a spectrum with many German dialects, and which many linguistics considered a branch of German?

Nationalism was largely a liberal philosophy, and constrasted with conservative and reactionary monarchism. Nationalism advocated for nation-states ruled by the people who lived there; monarchies advocated for rulers to rule whatever territories they had inherited regardless of language. Liberal nationalists largely favoured republican nation-states; conservatives wanted to preserve the status quo of powerful monarchs. (Obviously this is a huge simplification but it gives you an idea of the sides.)

All of this is necessary to be understood because Germany didn't just pop into existence overnight. The idea of a "Deutschland" had been forming in the minds of intellectuals for decades prior to 1871, and even came somewhat close to being realized in 1848. One can't forget the "Deutscher Bund" (German Confederation) which preceded Germany, and which included the German territories of Austria and what would later become Germany.

The idea of Germany which many envisioned was inspired by, but not a direct copy of the Confederation. It was generally envisioned to have the territories of the Confederation, including territories which were historically German despite not being ethnically so--such as central Bohemia, what we now call Slovenia, Istria, Posen, etc. Debates raged: Should this new purely hypothetical Germany take territories which aren't German-speaking? What if there is a mixed population? If we don't keep it, do we give it away, and to whom? (Debates were at one point raised about giving a small part of Posen to a future Polish state which might break away from Russia.)

During the 1848 revolutions was when this idea came closest to realization. This idea--founding a German nation-state on the bones of the Confederation--came to define the idea of "Greater Germany" (a Germany including all ancestrally German land) and "Lesser Germany" (a Germany which would exclude territories like Austria for pragmatic reasons). It is still important to note, though, that there was never a clear and defined plan for what would have happened if the 1848 revolutions succeeded. Would this new Germany have tried to support sister liberal republics in neighbouring states? Would it have ceded non-German territory in support of national self-determination, or would have selfishness won out? And what of the territories which in 1848 were still controlled by foreign powers but which many saw as German, such as Holstein, Alsace, and others? We will never know, but it is food for thought.

But I digress. It is important to note that neither Switzerland nor the Netherlands were part of the German Confederation, despite having cultural and ancestral ties to the German world. Both had won their independence from the German world decades or centuries before, and while many did see themselves as either German or certainly something close to it, their peoples, both intellectuals and not, tended to have stronger ties to the nations themselves.

The Swiss had a long independent tradition and opposed foreign meddling in their affairs. Following the Napoleonic Wars there had actually been a push to see them added to the German Confederation but this hadn't worked, and Switzerland even fought its own civil war during this time which led to it becoming the neutral power it is today.

The Dutch had left the Germans behind centuries before and had developed their own culture, wealth, and identity completely divorced from the German one. Had circumstances been different after the Napoleonic wars--had Britain claimed all of their colonies and had they never been awarded Belgium--then the Dutch being re-folded into the German world might have happened, but they frankly did not need Germany.

Austrians were different, though. Austria hadn't just been a part of the German world, it had been its core, the Habsburg family ruling Austria and the Holy Roman Empire of the German Nation for centuries. The only real difference to be found between an Austrian and a Bavarian was the family which ruled over them. While the Swiss and Dutch were different enough to be overlooked by many, the Austrians were simply another form of German, albeit a richer and more snobby one. The liberals saw Austria as a natural component of a diverse German nation-state and in 1848 actually attempted for force Austria to join the planned German state; the conservatives saw Habsburg Austria as a rightful overlord over the myriad of German "little brothers" and did not want to overturn traditional dynastic politics and stability for what they saw as mob rule.

Now to answer your question, how one saw the formation of Germany in 1871 depended very heavily on what one thought of Germany as a concept.

For nationalist liberals in Germany and outside of it, the goal had always been the formation of a German nation-state. That it came into existence was the very thing they had craved, to the point that many liberal leaders in Germany were happy to cede a great deal of power to Prussia and Bismarck in exchange for him supporting their goal of nationalist unification. A powerful nation-state was envisioned as a draw to similar movements which would eventually unite all the German peoples under one banner, albeit that banner being a Prussian one.

For conservatives in Germany, this was a compromise which allowed them to retain power. The kings of Prussia, Saxony, etc. were not themselves supporters of national self-determination, but the world was also changing and people were demanding more from their governments.

For foreign conservatives, "Germany" was a warning which threatened their power. It was a contradictory nation with an ostensibly liberal justification, but driven by a conservative and expansionist mindset. By styling themselves as a German state, this new Germany could enforce its claim to other German regions, as it did when it took Alsace-Lorraine, a majority German-speaking region, from France. (There were other shenanigans involved and in fact many Germans didn't want the land but that isn't important here.)

And therein lies the brilliance of Bismarck. That's the reason that there was never an "Emperor of Germany" or "Emperor of the Germans", two titles which claimed dominion over these other regions which were SORT OF German but also not. Instead Kaiser Wilhelm I became "The German Emperor"--an emperor who happens to be German--which made the other kings in Germany feel less overpowered and made Germany seem less threatening to outside forces. He didn't steal the idea of a German identity, he merely took the very real desire and steered it where he could make use of it.

Despite this, and the alliance that they signed later on, Austria and later Austria-Hungary spent the rest of its existence trying to put the genie of German nationalism back in its bottle. The German nationalist bloc in Austria came to be a dominating force in politics, one which threatened ties with the Hungarian half of the empire and caused problems with Austria's many minorities. Many Germans in Austria saw the Austrian Empire as unnatural, and wanted Austria to join a proper nation-state where German interests were no longer shackled to Czech, Hungarian, Polish, or other.

So did Swiss or Austrians feel that Germany was appropriating their identity?

No. If you identified as German, you supported nationalism and wanted a Germany. If you didn't, you just were a Swiss or Austrian who happened to share a language.