I wouldn't really call Russia's system fascist. It's definitely authoritarian, but I don't think it's really aiming for fascism.
I would. If it looks like a duck...
You can pick almost any of the existing
definitions of fascism and modern day Russia would score extremely high on most counts. I'll use the 14 general properties of a fascist ideology, listed by Umberto Eco here:
1. The cult of tradition. Check. Putin's attitude towards Russian culture and traditional values is extremely conservative and defensive against any kind of perceived threat. There is also an element of adulation of the "good old Soviet times", especially by the youth who never lived in Soviet Russia. The old people remember how it was, but even then some of them would prefer the old days (if only because their situation is even worse now).
2. The rejection of modernism. Check, partially. There is a certain... glorification of "The Russian Way" where existence consists of eat potato, drink vodka, and operate crude but somehow functional machinery.
3. The Cult of Action for Action's Sake. Check, partially. Russia has been seen acting (in Ukraine most notably) seemingly irrationally and with no apparent regard for consequences. With obvious consequences. And the people don't care because they see a strong leader doing things as better as doing nothing and being seen as weak.
4. Disagreement is Treason. Check, and how. Being a journalist is a hazardous proposition in modern Russia. Being an opposition politician even more so. Dissenting elements are easily branded as "foreign agents" which can be treated fairly harshly in terms of freedom of expression. Most recent example
here.
5. Fear of difference. Check. Russian population has a mile wide streak of intolerance, manifesting in racism, homophobia, xenophobia, and other unpleasantries, and this couples with their cult of tradition.
6. Appeal to a frustrated middle class. Check, of course, especially as the economic boom of the 2000s is abating and the international sanctions for Russia's actions abroad primarily end up affecting the middle class and the impoverished.
7. Obsession with a Plot. Check, see earlier example of an independent polling agency being declared a "foreign agent". The Chechens are also always plotting, as are the gays and atheists.
8. Portraying the enemies as simultaneously too strong and too weak. Partial check. United Russia's attitude towards opposition leaders is one example; they at the same time persecute them to the point of making them essentially political prisoners, suggesting they are too strong to be left unchecked, and implement restrictions on their presence in media to prevent fair elections from happening. At the same time, United Russia portrays them as decadent, corrupt, and too weak and feeble to ever make a difference against the popularity of United Russia. Another example could be made of their treatment of the LGBT community - they implement legal persecution, seemingly fearing them to be able to infect the youth with their "gay propaganda", but at the same time they basically dismiss them as not being worth engaging in dialogue with.
9. "Pacifism is Trafficking with the Enemy". Check. Being a strong military power is quite high in United Russia's (Putin's) priorities and ambitions, and certainly I could see them branding anyone arguing for a reduction of military spending to be a foreign agent, paid or otherwise contracted to reduce Russia's military capabilities.
10. Contempt for the Weak. Check. Ties to the cult of action for action's sake - anything is considered better than appearing weak. There is a large amount of contempt towards Yeltsin due to his apparent weakness, and by association, democracy is seen as weak. Not a good combination.
11. Everybody is educated to become a hero (cult of death and martyrdom). Not necessarily a check as such. Although Russian culture does promote a certain degree of "FOR THE MOTHER RUSSIA" -brand of martyrdom, by and large I suspect it's seen to be better to
live for the Mother Russia, than to
die for it. That said, the needs of the Motherland surpass the individual's right to live, and being ordered to die in battle without even a rifle to carry was a thing in WW2. The probability of that happening again is small as long as symmetric warfare is suppressed by the threat of mutual destruction, but some vestiges of that ideology probably still survive.
12. Machismo. Check. War and violence is idolized. Women are disdained. Nonstandard sexual habits are condemned. Doing manly man things is good, anything other is ridiculed. Putin without shirt doing manly man things is especially good, of course.
13. Selective populism. Check. United Russia portrays itself as the proponent of People's Common Will, but in actual fact it defines what that "Common Will" is supposed to be. And Russia being Russia, depressingly enough the people seem to voluntarily mold themselves to that Common Will, probably in fear of being ostracized, sanctioned, brutalized or killed.
14. Newspeak, or using fixed, limited, and fearmongering terminology in order to prevent people from applying critical reasoning to the terms used. "Foreign agent" is a term that's often repeated. "Gay propaganda" is another. There are undoubtedly more of these.
However, it should be noted that a lot of countries share many of these features. Notably, USA itself is actually quite similar in many respects, although the political system is not quite so openly and obviously corrupt and that flamboyantly disregarding of personal liberties - so far.
Also, these features being present doesn't mean the system is fascist. Some of these traits of fascism are about population being receptive to fascism, some of them are common tactics used by fascist ideologies - but these tactics are not necessarily
limited to fascism, other politically unscrupulous parties may employ them also.
That said, in the big picture Russian society appears remarkably prone to fascism, both in terms of the political system (practically autocratic president with almost unlimited executive power and a strong hold over the legislative and judicial branches of power), culture, and the population's general attitudes.