Fundamentally, because both chose to define themselves in this way.

Nazism was predicated on a worldview that posited that all harmful influences on German national life was the result of a Jewish conspiracy to undermine Germany's racial, cultural and economic strength. Removing this supposedly negative influence - through direct action, the legal system, deportation and ultimately extermination - was at the heart of Nazi policy priorities after taking power in 1933, as was strengthening the German racial community in other ways. According to Nazi ideology, which saw history as a struggle between races for supremacy, meaning that purifying and strengthening the race was essential in order to achieve victory over other races in the expected future struggle.

The most important manifestation of the Jewish enemy was "Judeo-Bolshevism", painting communism (alongside other forms of Marxist ideology, as well as other ideologies associated with modernity such as feminism) as part of a plot to destroy "Western" values, culture and society. This plot was global in scale, with adherents in every country, but was manifested most clearly in the USSR, where the Bolsheviks had actually taken power in 1917. The Soviet Union was therefore the main "other" against which Nazi Germany could define itself for most of its existence, and war in the east - to secure German living space as well as destroy the Judeo-Bolshevik enemy once and for all - was integral to Hitler's plans for the future, the defining struggle in which the German race would achieve its destiny.

For the Soviet Union's part, it was clear enough from the first manifestation of fascism in Italy in the early 1920s that it was an ideology that was fundamentally opposed to Marxist movements, that would seek to defeat them using its own violent mass mobilisation and, if successful, the power of the state itself. Marxist theories of fascism emerged over the interwar period, generally holding that fascism was the final stage of capitalism, where only open dictatorship and violent repression would suffice to stave off socialist revolution. Bulgarian communist Georgi Dimitrov provided perhaps the most famous formulation of this definition, that:

Fascism is the power of finance capital itself. It is the organization of terrorist vengeance against the working class and the revolutionary section of the peasantry and intelligentsia. In foreign policy, fascism is jingoism in its most brutal form, fomenting bestial hatred of other nations.

In the Marxist view, therefore, fascism was indeed the opposite of communism, the last ditch efforts of their opponents to stave off their defeat using whatever means we're necessary. This definition is not uncontested - it is not fully useful in explaining the particular dynamics of fascism as a movement - but as we're discussing the Soviet self perspective more than anything else, it's this definition that matters here. Both saw the other as their natural ideological enemy, and fully expected there to be a showdown (though, both sides could also be expedient about this - the Molotov-Ribbentrop Pact is proof enough of this, as it suited both sides' goals to play for time and concentrate elsewhere).

There is however another school of thought, external to the perceptions of either side, that holds that both Nazi Germany and Stalinist Russia were essentially similar, as both were totalitarian systems - whatever their ideological trappings, the regimes in practice were basically the same in their treatment of the state, party and individual. This analysis gained most credence during the early decades of the Cold War, when it was politically useful in the West to conflate the USSR with the defeated Nazi enemy - equally, it was useful in the Eastern bloc to conflate fascism with late-stage capitalism. Personally, while I find totalitarianism a useful concept when considering either regime, it acts to conceal real differences in the way each state was governed when used as a blanket form of comparison. To take just the most obvious example, state influence over the economy was constructed very differently in each state, from outright state ownership in the USSR, to the co-option of industrial capitalism in Nazi Germany (big business did very well out of cooperating with Nazi priorities, both in terms of war production contracts but also in terms of the "Aryanisation" of the German economy).