Back when this part of Europe was behind the Iron Curtain, the communist governments were quick to label any dissent moves as being fascist or nazi. This was their way to easily dehumanize their adversaries. Want freedom of opinion? Fascist. Want democracy and political pluralism? Fascist. Want true free speech (i.e. be able to say anything on your mind if it doesn't infringe the rights and liberties of other people)? Fascist.
The legacy of this behavior was continued after the fall of communism. I can only think of 2 examples: Romania and Transnistria.
Here in Romania, following allegations of voter fraud at the first free elections in the 1990s, large groups of people started protesting in big cities, including Bucharest. Ion Iliescu, the winner of the elections, claimed that the protesters were "huliganic elements, fascist elements, many of them under drugs" and called the miners from the Jiu Valley to come and restore order. The miners promptly came, went to the epicenter of the protests and beat people, injuring many, even killing some, burning cars, breaking windows and occupying the University of Bucharest. Basically, if they found you wearing a beard and glasses, you were fitted into the description of intelectuals and be shown a lesson.
Right across the Prut river, Transnistrian separatists also spreaded the propaganda that the Chișinău authorities were fascist and that they planned to unite with Romania without their agreement. Something that was not true.
This whole fascist/nazi talk seems to have been strongly inherited and integrated into the Russian culture of today, just like many things that were quite common during the Soviet period. So it's not so weird to see this rhetoric also being inherited and integrated as well.
In fact, this whole thing just means nothing. Nazism means simply nothing to them. It's just a word they use to describe whoever disagrees with them. Protesters? Nazi! Countries that are not Russia's puppets? Nazi. Why? because a former KGB officer gone president said so.
What's ironic is that Russia itself is objectively nazi. Yet they do not see the beam in their own eyes.