It's true that Putin clearly harkens back to the Soviet era in the sense that he imagines Russia as commanding a huge international sphere of influence and imperial ambitions. But there's also a minor irony in pulling down Soviet monuments, in that Putin's government is also virulently anti-communist. That aspect of the USSR he holds no nostalgia for, as an ultra-capitalist and Christian nationalist.
I feel like cold war narratives about the Soviet Union shaped our perception of its people. We tend to assume that everyone in the Soviet Union was on the political left, that they essentially believed (however hypocritically) in some kind of egalitarian society. The reality is that a lot of people in the Soviet Union subscribed to ideals and beliefs that we would associate very strongly with the political right.
Western media sometimes refers to Putin is a strongman, but as I understand it this is a literal translation of a Russian word,
silovik, that has a more specific meaning. Namely, it means someone with a background in the security services. These people essentially function as both a subculture and social class unto themselves in Russia. They have a particular in-group mentality that in many way resembles organized crime (indeed, there are a lot of overlaps between
siloviki and organized crime). Because the security services were deeply corrupt and nepotistic, many have family ties or histories with the security services going back multiple generations and have developed a kind of hereditary culture.
The people who joined organizations like the KGB were never idealists. They never saw themselves as building a better or more egalitarian world, but as an elite with the personal strength to do what was necessary to advance the national interest. They were typically social conservative and distrustful of anything and anyone which challenged the status quo. In any other society, they would have been called right-wing, and when the Soviet Union fell they didn't fundamentally change, they just became openly right wing.
While Westerners who feel nostalgic about the Soviet union tend to be leftists (at least on paper, I don't think that's as clear cut as tankies like to pretend it is) it's perfectly possible, in a Russian context, to be right wing while expressing nostalgia for the Soviet Union. I think a lot of people who lived in the Soviet Union always understood the performative nature of Soviet ideology in relation to the political reality. The nostalgia is for a strong, authoritarian state willing to enforce social conformity, clamp down on moral and intellectual degeneracy and compete with foreign powers on the world stage. For many people, that's what was "real" about the Soviet union, the ideology was just a tool to that end.
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