Why is Nazi Germany called a "fascist" regime?

To avoid admitting ideological closeness

Before we get to the problem of National-Socialists, let´s first address Fascists. It is a well known fact that Benito Mussolini was a Socialist before he started supporting Italian entry in WW1 in 1914. What is perhaps less known is that rump northern part of Italy, after the capitulation of Kingdom of Italy in 1943, was known as Italian Social Republic. Even the term Fasci (bundles in Italian ) was not entirely new invention in 1919 when Fasci Italiani di Combattimento was created. In fact, phenomenon of Fascist syndicalism existed even in 19th century. Of course, with fascism being a foreign word everywhere except in Italy, it was relatively easy for international Socialists to distance themselves from them and paint them as something completely opposite.

Now, to National-Socialists. It is quite unpleasant for international Socialists, that while being extremely national and racial, in economic policy National-Socialist were moderately leftists, something close to current Social-Democrats. In fact, they could have been even more close to communists in this regards had Gregor Strasser's faction gained upper hand in internal party struggle. At one point ranks of SA were overfilled with former communists. Hence the term Beefsteak Nazi. However, as it was, Hitler pursued mildly leftist policies during his rule, especially before the war. National-Socialists didn't expropriate factories and other property of bourgeoisie, like for example communists. But they did set up worker councils that participated in the management of various enterprises. They also increased benefits for workers, like for example paid vacation, better working conditions etc ...

Calling National-Socialists "Nazi" was good enough for Western propaganda, not so serious conversation/publishing and works of art (movies, songs, posters, placards etc .. ) . But, in its essence, "Nazi" is a nickname and a slur. West (US and Britain above others) were not that interested in explaining their ideological position towards Germany. In official proclamations they simply used term "German". For example: "The German High Command will at once issue orders to all German military, naval and air authorities and to all forces under German control to cease active operations at 2301 hours Central European time on 8th May 1945".

But Soviet Union and other international socialists and communist were in a bit of predicament. From their point of view, everything has to be explained in the light of "infallible" Marxist ideology and theory. Using cartoonish term like "Nazi" was simply out of question in serious works. And using the term National-Socialist would not only sow confusion, but could possibly give ammunition to those pointing eerie similarity between let´s say Soviet GULAG system and German concentration camps. Thus, blanket term fascist, although in its essence also connected with socialism, was used as more obscure to masses not knowing its true meaning.

Of course, in modern politically-correct times, term "fascist" remains to be used for serious allegations against someone, while term "Nazi" remains something for more low-brow conversation.

The simple reason is that Mussolini came first. He came to power in his famous March on Rome and subsequently became a bit of a role model for nationalist would-be dictators. When Hitler and Ludendorff tried to take power in Munich before a subsequent "March on Berlin" a little more than a year later, the parallels (beyond Blackshirts and Brownshirts) were already so obvious that comparisons between Mussolini and Hitler became quite common within Germany. E.g. von Lossows impression from 1924 that

He thought of himself as the German Mussolini, the German Gambetta, and his followers, who had taken on the inheritance of the byzantinism of monarchy, called him the German Messias

Er hielt sich für den deutschen Mussolini, den deutschen Gambetta, und seine Gefolgschaft, die das Erbe des Byzantinismus der Monarchie angetreten hatte, bezeichnete ihn als den deutschen Messias.

(quoted in Joachim Fest, Hitler, 2003 Berlin, p.292. But note that a contemporary newspaper has the quote without a mention of Mussolini on p. 2)

In fact, the comparisons were common enough to already back then attract objections from certain circles. E.g.

The German National Socialists are most often compared with the Italian Fascists, by friends and foes alike. Which is quite unjust. Fascism is morally and politically no better than German reactionaries, but its fundaments and development are completely different.

Die deutschen Nationalsozialisten werden von Freunden und Feinden meist mit den italienischen Fascisten verglichen. Ganz mit Unrecht. Der Fascismus ist moralisch und politisch nicht mehr wert als die deutsche Reaktion, aber in seinen Grundlagen und seiner Entwicklung ist er völlig verschieden von ihr.

(Weltbühne 2. Halbjahr 1924, p. 413)

For this reason it is ridiculous to talk about German fascism. An essential difference between Italian fascism and German secret society-ism is that Fascism has positive ideas, while Hitlerism has no ideas, that Fascism has constructive and Hitlerism destructive consequences.

(The term used for Hitlerism here is actually rather derogatory, but I have no idea how to properly translate it.)

Darum ist es auch lächerlich, von deutschen Fascisten zu reden. Der italienische Fascismus und die deutsche Geheimbündelei unterscheiden sich wesentlich dadurch, daß der Fascismus positive Ideen hat, während die Hitlerei ideenlos ist, daß der Fascismus konstruktiv, die Hitlerei destruktiv wirkt.

(Letter to the editors of Weltbühne from September(!) 1923. Weltbühne 2. Halbjahr 1923, p.252)

For a more complete picture, the National Socialists were not the only German organisation of the 1920s to be called fascist, other violent rightwing organisations were labeled similarly, e.g. in this article from early 1923.

This answer is a bit Germany-centric. For usage in English, I think the ultimative answer is again a mixture of "obvious parallels" and "Mussolini came first". Orwell's 1944 text in which he acknowledges that the term Fascist is applied to Germany a lot has already been mentioned. There is an earlier essay by Churchill from 1937 named "The infernal twins", where one twin is communism and the other one is fascism and national socialism. I am sure there are earlier examples.