How did they view themselves? Let's ask them.
Take the diaries of Joseph Goebbels. He writes in the extant works about national-socialism and fascism, mostly as a differentiation.
But differentiating between what? He does not identify himself as a fascist. The fascists are the Italians, under Mussolini. They are not only allies. They are brothers in mind and spirit.
He criticises fascists as being too lenient on the Jews, the fascists to be pitied, yet much closer to nazis than anything "liberal" (p891). He talks about fascists being something else yet the same.
In Goebbels mind, National-socialism and Fascism are one and the same 'thing', in broad conceptual terms. In his mind Spanish, English, Italian, Austrian fascists are the National-socialists of the respective country. Even identical twins can have slight idiosyncrasies in opinion or behaviour. They remain very closely related. But that is also a problem. Throughout these books it is not always easy to differentiate what he really means with 'fascists'. Quite some times he uses it just to stand in as a synonym for 'Italians', as that is apparently a word he seems to almost avoid until very late.
During his own rise nazism is just the German fascism, than the better fascism. When the luck of war turned, so does his view, identifying fascism now as just the current government of Italy:
Not only do the Italians do nothing in the field of warfare, but they also do hardly anything worth mentioning in the field of the arts. One could almost say that fascism had a sterilising effect on the creative life of the Italian people. It is not what National Socialism is. In contrast to its deep effect, it is a surface phenomenon. That is very regrettable, but we must be clear about it. National Socialism is in reality a world view. It starts all over again and lays new foundations for life. That is what makes our struggle so difficult, but also so beautiful ; and the goal that we will achieve in the process is actually worth the sweat of the best.
(9 Feb 1942, p1792)
The authoritarian sympathy is better exemplified in an earlier entry:
With Göring and d'Antinori into the deep night in a Russian cafe. D'Antinori told very interesting stories about fascism today. Mussolini does not seem to have recognized the Jewish question. Even in Italy, not everything that glitters is gold. But it has a leader, and he has the power. (13. Apr 1930, p477)
Source: Ralf Georg Reuth (ed): "Joseph Goebbels Tagebücher 1924 -1945", Piper: München, Zürich, 1999.
Instead finding mostly socialist and nationalistic sentiments. So did the german people under nazi rule openly identify as being fascist like Italy?
You have to keep in mind that 'socialist' sentiments were later in name only! If these sentiments were identical in actual meaning to what actual socialists, anarchists and communists – or even liberals (European and American meaning!) – understood that term to mean than the NSDAP would have to be called a wholesale fraud on that account alone. They redefined these terms to suit them.
The party programme of the NSDAP is quite slogan like in outlining actual plans and measures to be taken. The contents could be misidentified, and that was from the time that real socialist elements were driven out of the organisation, but kept with intention for working class appeal and brand recognition. That gave them some trouble for middle class appeal or pleasing their wealthy financiers when they still were trying to get votes.
But this was all explained in more detail eventually to illustrate their self-image and desired projected image, while demonstrating the un-socialists meaning of national-socialist:
Social and Socialist
"Yes, we call ourselves Socialist. That's the second step. The second step away from the middle class state. We call ourselves Socialist in protest against the lie of social middle class pity. We don't want pity, and we don't want social-mindedness. We don't care a hoot for that which you call 'social welfare legislation'. That's barely enough to kep body and soul together.
"We want the rights to which nature and the law entitle us.
"We want our full share of what Heaven and of the returns from our physical and mental labors.
"And that' Socialism!
Nationalist and Socialist
!Then we will prove that nationalism is more than a comfortable moral theology of middle class wealth and Capitalist profit. The cesspool of corruption and depravity will then yield to new nationalism as a radical form of national self-defense, and to new Socialism as the most conscious creation of its requisite preconditions.
This goes on to smear Marxism, Jews, Monarchists, Republicans, Internationalism, Capitalism or Parliaments, Pacifism, and solidarity, while at the same time outlining a still very capitalist corporatist state. These are not elements of socialism as it was understood then or understood now.
"Marxism will die, so that Nationalism may live! And then we will shape the new Germany – the nationalistic Socialist Third Reich!"
From the very handy Joseph Goebbels: "The Nazi-Sozi. Fragen und Antworten für den Nationalsozialisten", Verlag der nationalsozialistischen Briefe: Elberfeld, 1927/1932. (On archive.org)
Another prominent Nazi wrote this, but note the name and the date:
We therefore felt that the republican-monarchist semi-darkness deliberately maintained by the party leadership was a burden, the exaggerated reverence for the fascist authoritarian state, which is becoming more and more apparent on the part of the official party authorities, almost a danger to the movement and a sin against the idea.
Source: Aufruf der Otto-Strasser-Gruppe vom 4. Juli 1930: »Die Sozialisten verlassen die NSDAP« (The socialist are leaving the NSDAP)
The well known painter with moustache talks extensively about Italian fascism in that book he is known for. He praises most core ideas, criticises what he thinks more important or underdeveloped in fascism ("the Jews!"), but not once does he call himself in that way a fascist.
And that illustrates other problems with the label "fascist":
on the one hand it was seen as 'an Italian thing', on the other hand it was of little use as a clear differentiator to other right-wing parties and organisations, who were almost all labeled 'fascist', somehow. At once by the communists of course.
But at the same time by themselves then, or the conservatives and other radicals at least ventilated the greatest sympathy, admiration and model character of fascism. When the term came to be known in Germany, it at first did not have that much of a negative connotation in itself.
The Stahlhelm, the Black Reichswehr, the DNVP for example did so.
Examples from Hugenberg's sphere: Ludwig Bernhard, Hans Meydenbauer: "Faschistische Eindrücke", in: Preußische Jahrbücher Vol. 201, Juli 1925, H. 1, p105-109; "Faschistischer Fortschritt", ib., Vol. 202, November 1925, p271-275. // Reinhold Quaatz: „Italienischer Nationalismus“, in: Das Freie Deutschland. Nationale Zeitschrift für Politik und Wirtschaft, Nr. 8 vom 21.11.1931. These examples are clear indicators, but limited in time, as those right-wingers later viewed fascism and then national-socialism as still "too socialist" for their tastes.
To stand apart the nazis were well-advised not to overplay the similarities, even in names for self-description, and in terms generally thrown around:
In the second half of the Weimar Republic, several articles appeared in the right-wing journalism which dealt with the transferability of the Italian model under the heading "German Fascism". In April 1927 Helmut Franke made the start with a three-part series of articles in Arminius, to which Heinrich von Gleichen answered already in the following month. Three years later Max von Binzer put the matter on the agenda again in the Deutsches Adelsblatt (German magazine for the nobility, after he had already in 1928 in the yearbook of the Centre International d'Etudes sur le Fascisme, published in Lausanne, identified parallels to fascism among the Alldeutschen (All-Germans), the Deutschnationalen (German Nationals), in conservatism and in the Stahlhelm (steel helmet). Another article by Willy Hellpach appeared in the special issue of European Revue on the topic "Ten Years of Fascism" in November 1932. (Links added for convenience)
Maurizio Bach & Stefan Breuer: "Faschismus als Bewegung und Regime. Italien und Deutschland im Vergleich", VS: Wiesbaden, 2010.
The label with the word "socialist" being in the very name of the party?
Obviously, there was no humanitarian impulse or desire for a new form of society in Hitler’s version of socialism. He himself declared that his socialism had “nothing at all to do with a mechanical construction of economic life”; rather, it was the complementary concept to the word “nationalism.” Socialism meant the responsibility of the whole for the individual, whereas “nationalism” was the devotion of the individual to the whole; thus the two elements could be combined in National Socialism. This prestidigitation allowed all interest groups to have their way and reduced the ideas to mere counters: capitalism found its true and ultimate fulfillment in Hitler’s socialism, whereas socialism was only attainable under the capitalistic economic system. This ideology took a leftist label chiefly for tactical reasons. It demanded, within the party and within the state, a powerful system of rule that would exercise unchallenged leadership over the “great mass of the anonymous.” And whatever premises the party may have started with, by 1930 Hitler’s party was “socialist” only to take advantage of the emotional value of the word, and a “workers’ party” in order to lure the most energetic social force. As with Hitler’s protestations of belief in tradition, in conservative values, or in Christianity, the socialist slogans were merely movable ideological props to serve as camouflage and confuse the enemy. They could be changed or rearranged, depending on the situation. The leaders, at any rate, were totally cynical about the principles of the program—as one enthusiastic young convert learned from a talk with Goebbels. When the young man remarked that Feder’s call for smashing the enslaving system of interest payment did contain an element of socialism, Goebbels replied that what ought to be smashed was anyone who listened to such twaddle.
Joachim C Fest: "Hitler", 805.6 (eBook), Harcourt: New York, 1974.
Also noteworthy is that many conservatives, authoritarian right-wingers, interpreted fascism as the Prussification of Italy.
Authority, discipline, military, brutality, warmongering, ant-democratic tendencies. What's not to like about that? (Given the 'right' mindset.)
When Hitler was given power, the German conservatives were already fascists in mind and heart. That was the perfect soil on which national-socialism could sprout. In such climate a communist might think 'fascist' would be an insult, but right-wingers back then would just emphasise the small details of difference and otherwise just think "Yeah, so what? Brown is the new black."