Four the most part, no, neither the USSR's leadership nor the Third Reich's really see "socialism" or "national socialism" mirrored in FDR's New Deal. The important thing to realize though is that "socialism" meant different things in both dictatorships. But even by these distinct metrics, the New Deal fell short of them.

The Soviet case in rejecting the New Deal was more thought-out and comprehensive than the Third Reich's. This was because the USSR had an ideology- Marxist-Leninism- that had a clear definition of what socialism was and how to achieve it. In Marxist-Leninist orthodoxy, socialism was the indeterminate stage between a revolution and the communist utopia predicted by Marx where man would want for nothing. Socialism would come through the state control over the economy but the state would in turn be controlled by the workers through the mechanism of a communist party. So for there to be real socialism for a Marxist-Leninist, not only did there have to be a revolution but such a revolution had to be led by a vanguardist and disciplined communist party.

This formula was one of the reasons why Soviet leaders and propaganda organs tended to dismiss the New Deal. Not only was it a byproduct of the establishment American political system, it did not fundamentally alter the structures of American society. The 1979 entry on the New Deal in the Great Soviet Encyclopedia is an instructive look at this dismissive attitude:

New Deal

a system of measures undertaken by the US government between 1933 and 1938 to mitigate the contradictions of American capitalism, which had been aggravated as a result of the economic crisis of 1929–33. The New Deal is associated with President F. D. Roosevelt.

During the first period of the New Deal (1933–34) laws advantageous to large-scale employers were passed. The National Industrial Recovery Act (NIRA), which was passed in 1933, provided for the introduction into various branches of industry of “codes of fair competition,” which fixed production costs and levels of output and allocated market areas. Basically, the codes supported the largest monopolies at the expense of small and middle-level employers. The Agricultural Adjustment Act (AAA), which was passed in 1933, was designed to raise prices for farm products by paying monetary compensation to farmers who reduced sown areas and livestock herds. These and subsequent New Deal economic measures directed at state regulation of the economy constituted an important stage in the development of state-monopoly capitalism in the USA.

A number of laws were passed during the second period of the New Deal (1935–38), a time marked by the growth of the working-class and democratic movements: Among those passed in 1935 were the National Labor Relations Act (the Wagner Act), which reinforced NIRA provisions concerning the right of workers to organize trade unions and conclude collective agreements, and the law providing for social security and for aid to the unemployed, the first such law in the history of the USA. The Fair Labor Standards Act (1938) established minimum wages and the maximum length of the workday for certain categories of workers.

The labor and social laws of the New Deal expressed the government’s aim to deaden the class struggle and weaken the workers’ and mass democratic movement {emphasis mine}. Through their struggle to expand the framework of bourgeois democracy, the toiling masses forced the ruling circles of the USA to undertake reforms and compromises.

Granted, the GSE entry was written over forty years after the New Deal and during the Brezhnevian neo-Stalinist period, but it does outline the basic objections a doctrinaire Marxist-Leninist would have over the New Deal. One, state regulation was done in cooperation with moneyed capital interests. FDR was not seeking to change socioeconomic relations or concentrations of wealth. Second, the New Deal's programs could not alter the fundamental logic of historical development in which capitalism's contradictions created the very conditions for a revolution. In this Marxist-Leninist orthodoxy, the solutions to these contradictions were often the cause for the next stage of the dialectic. Finally, and related to the second point, the New Deal sought to reform capitalism which in Marxism is unreformable.

Now there was some tactical embracing of elements of the New Deal during the 1930s. Soviet agitprop would occasionally commend FDR for working with "progressive forces" within the US such as grass-roots unions. This positive line was more often associated with the "Popular Front" era of the early and mid-1930s of global communism where communist parties made common cause with their "bourgeois" counterparts in the left to fight the emerging fascist threat. But there was a Janus-faced element to this view of the US where some Soviet commentators found the New Deal a species of fascism. The CPUSA in 1933 declared the New Deal to be "a program of fascistization and the most intense preparations for imperialist war." Ilf and Petrov's famous 1935 road trip across America interspersed between celebrations of America's "can-do" attitude and the examples of grinding poverty and close-mindedness evident in the darker corners of US society. Stalin in private would argue that the New Deal's successes were always incomplete compared to the Soviet 5YPs which were unburdened by bourgeois norms. Stalin himself could alternate between Popular Front-like overtures to FDR and cold dismissal of America's efforts to alleviate the Depression. But events within America were generally pretty marginal within Soviet discourses in the 1930s; events in Europe and East Asia were more directly concerned with Soviet security than FDR's "Alphabet soup" of government programs.

Stalin's general ambivalence to the US was mirrored in German estimations of the New Deal. The NSDAP press displayed the same tendencies as the USSR's to either celebrate the New Deal as a pale reflection of Hitler's domestic program or condemn it as a pitiful attempt to achieve German successes without the fundamental catalysts of a strong political party or a vital leader like Hitler. The more positive estimations of the US tended to cluster in the early stages of the Nazi seizure of power- ca. 1933-35- when the diplomatic relationship between Germany and the US was relatively cordial. The Nazi press did comment favorably on the New Deal's handling of the banks and that the CCC was proof that the Americans were copying the Arbeitsdienst (which itself was a Weimar era program that the NSDAP claimed as its own and militarized) and other German relief programs. But the growth of American Jewish boycotts as well as US reporting on Nazi violence compelled a more negative assessment of American political developments in the German press. Hitler would maintain in private conversations the few times the New Deal came up that it was emblematic of a Jewish-dominated polity and it lacked the genuine racial character of National Socialism.

In a curious parallel to some of the Soviets' praising of the New Deal as an American Popular Front, the Germans sought to cultivate anti-New Dealers as US policy towards Germany became more hostile under FDR. Hitler granted Herbert Hoover a private audience in March 1938 and the German dictator not only noted that the National Socialist experiment was different in character than the New Deal, but that anti-New Dealers shared common ground with the NSDAP on account of their anti-communism. German envoys to the US also cultivated contacts within the isolationist wing of the Republican Party as well as the Army, which in turn made it appear as if the New Deal was more unpopular than it was and there was little for Germany to gain by lauding it.

The German courtship of anti-New Dealers bore only modest fruit as it strengthened what was already a strong resistance to FDR's initiatives. But as early as 1933/34, it was already becoming apparent among anti-New Dealers that likening the New Deal to Hitler's Germany was a viable way to attack the American program as a claw of governmental dictatorship. There were several press commentaries such as those in the conservative New York Herald Tribune that likened the NRA's "Blue Eagle" program to the proliferation of NSDAP paraphernalia like swastikas. Other 1930s commentators likened the Blue Eagle to the Winterhilfswerk (Winter Relief) which used voluntary charity by individuals and businesses who in turn received a badge showing their national solidarity. There were also concerns even within the New Deal coalition that the programs were imbibing German antidemocratic methods. One of the criticisms of the CCC was that the involvement of the Army and the issuance of uniforms made it too much like the Nazi Arbeitsdienst for an American republic.