The Axis phrase was popularized by Mussolini. In a famous 1936 speech in Milan, the Italian dictator commented on the recently-completed bilateral treaty with Germany

this agreement, which has been fixed in special Procès-verbal, duly signed this vertical line between Rome and Berlin, is not a dividing-line but rather an axis around which all European states animated by the will to cooperation and peace can collaborate.

Mussolini's framing of the treaty not only pushed back against larger notions that Germany and Italy were dividing the world, but also supposed that these two countries were refashioning the lines of power around the globe. An axis is a pretty neutral term on its own- it refers to an axis of rotation- but it does have a more laden meaning when applied to geopolitics. Both Hitler and Mussolini presented themselves as the champions of the countries that were have-nots and surrounded by hostile forces. Mussolini's quote above does assert that both Italy and Germany are not only breaking out of their cordon, but also fundamentally restructuring the balance of power. A geographic axis is a permanent thing and was a metaphor for fascist ideologues' tendency to see their moment as crafting a new geopolitical order. Hitler would refer to the Italian-German relationship after the Munich Pact as an Axis between two peoples, not a mere alliance.

Mussolini's assertions to the contrary, the alliance between Italy and Germany was hardly functional and had little interest in peace. But fascism's language tried to project images of strength that would be transformational. The Pact of Steel was one example of this amalgamation of puffery and menace. Italy and Germany had not only formed an alliance, but one that was as hard and strong as steel. So in this sense, Axis did have a sinister gloss to it; there was a good deal of rhetorical intimidation implicit within fascist rhetoric. By Mussolini's own words it was an alliance predicated on transfiguring the world along fascist lines. If you were ideologically opposed to fascism, then you were on the exterior of this new political axis.

The phrase Axis picked up some currency in the Western press as it described the connection between Rome and Berlin. A number of Anglo-American papers referred to the "Rome-Berlin Axis" when discussing the two dictators' actions within Europe as did some diplomatic correspondence, such as this report produced by the US State Department speaking of a "Rome-Berlin Axis" in Europe. Although it is far from scientific, this google ngram for "Rome-Berlin Axis" does show the term picked up speed in the late 1930s and crested when Mussolini joined the war in the spring of 1940. Japan's hitching of its star to Germany and Italy also helped popularize the phrase "Rome-Berlin-Tokyo Axis," although this was not as common as its earlier incarnation or the simplified term Axis.

So Allied discourses, ranging from news reports on the war to propaganda, helped further boost the visibility of the word Axis. For their part, the Germans tended to eschew the phrase Axis in their propaganda and discussions of the war's geopolitical alliances. Confusingly, the Germans sometimes used alliierte (allies) when speaking of their Japanese, Italian, and other comrades. Occasionally some broadcasts and materials referred to the Axis, but Nazi propaganda tended to cast its motley wartime alliance as the banding together of a civilized Europe against the tides of Anglo-American mammonism and Soviet Bolshevism, both of which were created by Jews. Much of Germany's propaganda tended to emphasize its war as a crusade against Jewish Bolshevism such as this recruiting poster for the Belgian SS or this French poster using the imagery of the Crusades. Naturally, Germany was the born leader of a cultured Europe as visualized by this striking 1941 poster. But the moniker Axis tended to stick much easier than a Nazi-led Europa.