There are really so many components to this question that I am unsure I can do them justice. I will surely try.

First of all, the "Axis" was made up of three primary countries: Germany (with attached Austria), Italy, and Japan. Japan's involvement was really more diplomatic with no Japanese soldiers fighting in the European theatre and no German soldiers fighting in the pacific. This alliance was mostly one of convenience with the different countries not necessarily have the same goals. Italy and Germany were allied because of ideology, Japan joined the Axis because they had few options.

It is worth noting that there was a difference between what a country stated its goals were and how they acted. Italy for example never stated it wanted world domination but Mussolini's foreign policy was indeed nationalistic and imperialistic - he was very interested in conquering what he considered traditional Italian territory (currently French) and expand the Italian sphere of influence. Japan also never professed an interest in world conquest but acted imperialistic in Korea and China.
The big question is, what was Hitler's goal. Did he want to conquer the world or just part of it? Germany professed to "just wanting lebensraum" meaning Hitler wanted to grow and expand the German people and needed space to do so. Hitler SEEMED to be content with conquering that space in the east, first by taking what he considered German territory in Czechoslovakia and Poland and expanding that into the Soviet Union. It is worth noting that he occupied western territory in France but he did not annex it with the exception of what he considered German territory in Alsace and Lorraine. It is unclear what would have happened had Germany won. Hitler clearly wanted a new world order with Germany at the center and vassal states surrounding him but it is not clear that he would have conquered them.

On the flip side, the Allies, initially France and England, later England, the Free French, America and the commonwealth nations, had no real need to pretend Germany was going to conquer them. France and England were under attack and got men to fight on that reasoning. America started the draft before they entered the war in September 1940. Propaganda in America was less to get more men to fight and more to keep the civilian population that did not see the devastation of the war supportive of the war effort.