The Soviet Union tried extensively to forge some kind of a Soviet "new man" identity, trying to tie together the different ethnic groups of the Union together. However, as 1991 demonstrated, this was a failure and different nationalities went their own way. In the end, Russians in Moscow did not feel as though a Balt was the same ethnicity as them (and vice-versa).
However, if you look at the 19th century, you see western European nationhoods being forged from disparate ethnic and linguistic groups all the time.
France in 1789 was a mish-mash of ethnic groups from Celtic Britanny to Occitians to Cosmopolitian Parisians. Italy similarly created an Italian national identity when in 1860 someone who lived in Naples or Sicily would have viewed a Piedmontese or Milanese as foreign as a Frenchman, and what was the become the Italian official language was not spoken outside a small percentage of people on the peninsular. However, governments in both countries successfully created a national identity which subsumed regional ones in the 19th century despite those challenges.
So what was it about the success of those nationhood building projects and the failure of the Soviet one?
The Bolsheviks established the Soviet Union on the ashes of the Russian empire, and needed a way to justify holding on to the old pieces of the empire that wanted to break away. Poland and Finland as well as the Baltic states had already broken free, and the Red Army had to step in to reconquer Ukraine and the Caucasus after they had declared independence.
As the nationalist aspirations of the non-Russian former subjects of the Russian empire could not be ignored, the entire structure of the Soviet Union was set up as a union of nominally equal Soviet republics representing the different nationalities in order to appease them. In the early Soviet Union, expressions of the non-Russian cultures were even encouraged at the expense of Russian "chauvinism" and the rhetoric was one of "friendship of peoples", not the creation of a single Soviet nationality. The early nationality policy of the Soviet Union which lasted up to the Great Terror of 1938-39 is described in The Affirmative Action Empire: Nations and Nationalism in the Soviet Union, 1923-1939 by Terry Martin.
Even after the Soviet state turned to suppressing the ethnic minorities of the empire, most vividly demonstrated in the deportations of entire peoples, the institutional structure of the Soviet Union that was conceived as a cooperation between different nations meant that coalescing around a single national identity would be difficult. It didn't help either that the Soviet Union acquired new territories at the end of World War II populated by non-Russians who had been able to pursue their own national projects in the interwar period (the Baltic states, Western Ukraine). And perhaps ironically, the growth of Russian nationalism also ran counter to the creation of an all-encompassing Soviet identity, due to the continued distrust of Russian "chauvinism" by non-Russians. Soviet leaders after Stalin from Khrushchev to Gorbachev were rather clumsy in promoting Mother Russia without paying enough attention to the simmering discontent of the non-Russians. I don't think I would agree that they seriously pursued the creation of a single Soviet identity—for them, the Great Russian identity was enough to represent the Soviet Union, and the non-Russians, especially non-Slavs, didn't matter much. Gorbachev's dismissal of an ethnic Kazakh from the position of First Secretary of the Communist Party of Kazakhstan and the subsequent appointment of an outsider from the Russian SFSR in 1986 that sparked widespread protests exemplifies this cluelessness of the Soviet leadership.
For all the attention (rightly) paid to the 'national minorities' since 1991, I think you need to step outside the national framework to answer this question.
The French drive to 'turn peasants into Frenchmen', in many ways a successor of earlier North American francisation efforts, relied on adoption of language and culture to 'become French'. For all the claims of 'civic nationalism', 'Frenchness' did/does demand an acceptance of the cultural and linguistic norms that originated in the Paris area.
The equivalent of this in the USSR would have been Russification. By the early 1920s this was ideologically unacceptable to the Bolshevik leadership. For all the flexibility of his conception of self-determination, one of Lenin's more admirable traits was his visceral dislike of 'Great Russian chauvinism'. Hence the policy of korenizatsiya - encouraging (or creating) ethnic languages and culture amongst the national minorities.
Now this would later be seen as a ticking time-bomb at the heart of the USSR but this can be misleading. Early Soviet leaders saw no contradiction between this (almost anti-francisation) policy and the creation of a New Soviet Man (NSM) because the latter had no national attributes.
That is, there was no difference between an worker who spoke Russian and one who spoke Armenian: both could be considered examples of NSM. Unlike France or Italy, the question of nationalism and modernity were not considered one and the same in the USSR. One could, in theory, quite easily be both Latvian and Soviet.
Rather than depending on an ethnic, cultural or linguistic heritage, the early Soviet theorists, as good Marxists, saw their New Man as emerging from the place of production. Socialist man was to be created from a socialist workplace - there was every confidence that new technologies and methods (eg Taylorism), and later sheer willpower, would shape the emerging Soviet citizen. Our NSM was therefore to be voluntarist, collectivist, heroic, politically conscious and loyal. Above all, NSM was new. As Gorky put it the "New Man is… young, not only biologically but also historically. He is a force that has only begun to become aware of his path and significance in history."
As you can probably see, by the 1930s the ideals of NSM had become entwined in the self-consciously heroic drive of Stalinist industrialisation. The new man could be found amongst the ranks of Stakhanovites, shock workers, artic explorers, etc. These were the role-models for a new Soviet identity and they were not bounded by ethnic/national characteristics.
So I would argue that in order to understand why the creation/adoption of NSM failed, we need to look at the failure of the socialist experiment itself. The utopianism of the 1920s and 1930s was dead long before the Union itself dissolved or nationalist consciousness came to the fore. For this I'd posit two key reasons:
It was inherently utopian. That is not criticism in itself but even in the 1930s, the ideology of NSM was always bumping up against the grinding reality of Stalinist industrialisation. The ideal was a powerful component of Soviet power but for every individual striving to improve themselves in the blast furnace of Stalin's revolution there were countless just struggling to survive. These were hard years and the default reaction of most workers was not to embrace shock work but to circumvent or resist the state's call for sacrifice. To combine Eugen Weber and Moshe Lewin, you could even argue that the mass migrations that accompanied industrialisation 'turned Soviets into peasants'.
This tension between ideology and reality was decisively settled in the post-war period under what Vera Dunham has called 'the Big Deal'. This was the post-Stalin settlement by which the Soviet state promised material comforts to its 'middle class' in exchange for refraining from political opposition. Implicit within this deal was a retreat from ideology, or at least relaxing the requirement for ideological commitment (something furthered by Khrushchev's Secret Speech). In the place of 'socialist consciousness', more stifling cultural criteria came to prevail and propaganda images of material abundance, while now closer to reality, lost much of their explicit political connotations.
The result of the above, plus no doubt many other factors, was that by the 1970s (the 'Little Deal' of the Brezhnev years) the drive to create a New Man had largely petrified. The political spirit that was meant to animate him was effectively dead; Gorky's ideal of a young thrusting class had given way to a gerontocracy and its largely apathetic population. It was in this period that the term Homo Sovieticus was coined as a cynical response to the idealogical void at the heart of Soviet state and society. Attempts to build a NSM and instil a socialist consciousness in the population had failed.
Hence my reference to nationalism as a time-bomb above. It became explosive when the Soviet experiment itself failed - ethnicity became an alternative pole of identity when the Soviet system itself was clearly breaking down. It was the absence of a Soviet identity that allowed nationalism to flourish. But, in my view, the quest to forge a New Soviet Man had broken down long before this. Its failure needs to be traced to the slow bankruptcy of the Soviet system that was supposed to give birth to it.
The French could go to the countryside with a national identity based on already existing ethno-linguistic standards, the Soviets had to create a new identity, a new man, from scratch. But the reality of the Soviet experiment left this an unrealised utopian dream.
Sources
A few key works referenced above:
Jochen Hellbeck has done a lot of work on this, which I don't always agree with. However his joint essay (The New Man in Stalinist Russia and Nazi Germany) in Beyond Totalitarianism on the subject. That's the source of the Gorky quote.
On the tension between ideal and reality under Stalin, see Moshe Lewin's Making of the Soviet System, Don Filtzer's Soviet Workers and Stalinist Industrialisation and Stephen Kotkin's Magnetic Mountain. Filtzer's work is particularly useful in looking at how, far from building a socialist workforce, the ideals propagated from above led to semi-anarchy on the shop floor.
Vera Dunham's In Stalin's Time is a classic framework that I quite like. The 'Little Deal' and subsequent overwhelming cynicism in Soviet public life is the subject of James Millar's Brezhnev's Contribution to Acquisitive Socialism
For Italy we can definitely discuss about their success of forging a nation. The conflict between North and South, which often took the form of raciscm and discrimination, is an ongoing one, to this day. While they succeeded in producing a single language spoken by all Italians, culture, attitude and politics are still very divisive between North and South. The splitting of the country isn't mainstream but it also isn't a super fringe topic. I'd argue the thought that Italy has a real National identity and a shared historic memory.
And for France we can argue time. You can argue that the process of frenchification started in the 13th century when Occitan culture was supressed and attacked during the Albigensan crusades. This didn't happen with a plan or with the intent of forcing a French culture. But the losers of the war and their culture was supressed. In the 19th century this process then became intentional and with a clear aim. The Soviet Union didn't even exist for 100 years during a time where Nationalism was the strongest, most important and extensive ideology.