Nazi Germany expected that a swift blow would cause an effective collapse in Soviet resistance, forcing the country to sue for peace in months. From answers such as this opinions on Soviet capabilities were similar among the allies.
I understand that the USSR's industrial capacity and manpower were enormous compares to everyone else in Europe, which is why this is a political and psychological question. After the German breakthrough to the French coast, France suffered a catastrophic loss of morale and was totally unwilling, for instance, to fight on from Africa.
The USSR's circumstances were quite different, but there were significant similarities in the millions of encircled men and the rapid bypassing of its border fortifications. Nonetheless, their troops fought on when surrounded far longer than France's, and there was only the barest hint of political unrest. The Chechnya Insurgency was tiny compared to the USSR's scale, even though the disastrous preparation for the conflict would made something as extreme as deposing Stalin feel totally unsurprising.
I can think of a few possible reasons for Soviet political resilience to exceed France's, such as the USSR's lack of free press, it's indoctrinated people, and west Europeans failing to grasp the geographic scale at which Russians are forced to think at, but don't have any basis for thinking it's any in particular.
There are a couple of incorrect premises that you're operating on that make it a bit difficult to offer a direct and simple answer.
Nonetheless, their troops fought on when surrounded far longer than France's, and there was only the barest hint of political unrest. The Chechnya Insurgency was tiny compared to the USSR's scale
First and foremost, there was a significant amount political unrest depending on how you define political unrest. No, Stalin's position and that of the Soviet leadership wasn't in trouble of being overthrown from within, but there were widespread reports of German's being welcomed as liberators in certain regions of Soviet controlled areas and the USSR proper, up to and including outright collaboration and the formation of volunteer units fighting alongside the Nazis. I don't specifically know how you're defining political unrest, but that active collaboration of an invading force fits with most definitions I have encountered.
Secondly:
such as the USSR's lack of free press, it's indoctrinated people
This leaves aside the very real and objective reasons why a Soviet citizens might want to rally against the Nazis, in particular when facing down one of the largest military operations in human history. On the one hand, if a Soviet citizen was accurately informed about the political situation in Nazi Germany they would understand the ideological and racial animus toward the USSR that underpinned Nazi ideology. 'Judeobolshevism' was a constant boogieman invoked by the Nazis, directly tying their rampant antisemitism together with their ever present anti-communism. War was, at a certain point, viewed as essentially inevitable. To quote Hiroaki Kuromiya in his biography of Stalin: "No one, including Stalin, knew when and how war would begin, though nearly everyone in the Soviet Union, including Stalin, knew that war would come." [pg. 133 of Stalin by Kuromiya]
Further, regardless of what Soviet citizens thought of (or even knew about) the Nazis at the beginning of the conflict, events made it abundantly clear very early on what the trajectory of the invasion would be. Mass killings of communist political officers and Jewish fighting age men morphed into wholesale slaughter, ethnic cleansing, and eventually the Holocaust. Rural villages were burned and their inhabitants shot. Einsatzgruppen were unleashed upon the lands the Nazis had conquered. Before the invasion, German soldiers were given these directives in the form of the 'Guidelines for the Conduct of the Troops in Russia':
Bolshevism is the deadly enemy of the National-Socialist German people. Germany's struggle is against this disintegrative ideology and its carriers. This struggle demands ruthless and energetic action against Bolshevik agitators, partisans, saboteurs, Jews - and the radical elimination of all active and passive resistance." ["War in the East and the Extermination of the Jews" by Hillgruber]
To be perfectly blunt: propaganda, censorship, and misinformation aren't necessary when you're the target of a war of extermination. For millions of people under the who found themselves confronting the horrors of the Nazi invasion, the reality before their eyes was more than enough to inspire resistance.
There are, of course, other answers to this question, but hopefully this clarifies a bit of the context for thinking about WWII from the Soviet perspective and gives you a good frame of reference for some of the other responses you get.
Edit: whoops hit enter as I was in the middle of typing my final sentence.
Final point to keep in mind as a point of comparison between the two regions you brought up: the French Resistance during WWII was largely made up of socialists, communists, and anarchists, as they had a direct ideological opposition to Nazism. Not everyone France was opposed to the Nazis in principle (and as I pointed out above, the same is true of the USSR). In the same way, if you identified as a committed socialist or communist in the USSR (a self-described socialist country), then the odds were good that you would also take a fascist invasion as a rallying call to resistance. Reactionary nationalists and socialists/communists tended to be historical enemies, as recent civil conflicts in both Germany and Italy during the rise of their respective fascist movements made abundantly clear.