As you note, fighting (or more generally, violence) has been a part of North American hockey since pretty much the first game was played in the late 19th century. Stories date back to then about players being bashed on the head (among other regions) with sticks, needing to leave the ice for some stiches on open wounds, and returning to play. There's also reports of more serious incidents, including at least one death directly caused by on-ice violence in the early era (the death of Owen "Bud" McCourt, who was hit in the head several times with sticks during a March, 1907 game, and died the next day; one player, Charles Masson, was actually charged, but not convicted as so many players hit McCourt it could not be determined who struck the fatal blow). This is all said to give an idea of how the sport developed, and that it's never really lost it's physical nature (in North America at least).
Hockey was for the first few decades mainly a North American-exclusive affair. It was only around the turn of the century that it really picked up in Europe, and that was due to the influence of Canadians living over there. A major source of this was actually thanks to the Rhodes Scholarship program at Oxford: Canadians studying there effectively introduced the game (similar sports had been played before), and toured Europe, specifically France and Switzerland, bringing the game there. Indeed the first Canadian national team, if one wants to call it that, played at the 1910 European Championship (unofficially), and was simply the Oxford Canadians, as they were known. This team would continue to tour Europe for several years, and in 1923 won the first Spengler Cup (a now annual competition held in Switzerland in late December), with one of the players that year being Lester B. Pearson, the future Prime Minister of Canada.
From the start of the Canadian tours two things were readily apparent to the European audiences: how much better the Canadians were than the locals, and how much more physical they were. Unlike the North American game (really, Canadian; the American university system banned bodychecking until the 1960s), the Europeans simply didn't partake in the violence, even restricting bodychecking: when the LIHG ("Ligue Internationale de Hockey sur Glace", the forerunner to the IIHF, the International Ice Hockey Federation, the governing body of nearly all hockey worldwide) was formed in 1910, it explicitly banned bodychecking in sanctioned games; as the LIHG was a Eurocentric body, this meant European matches had none of that. This was relaxed in 1926, when it was allowed in the 1926 European Championships, and only allowed in all zones starting in 1969. With such a restriction, I think it goes without saying that fighting was explicitly banned, and not tolerated at all (even today IIHF matches have players who fight serve a one-game suspension).
As for the Soviet hockey system in particular, they followed a slightly different path. While the sport was known to an extent in the late Russian Empire and early Soviet Union (as Canadian hockey), like most sports it was not played on a national level, or really at all. The Soviets discouraged sports at this time as a bourgeois pursuit (not entirely without merit; the development of modern sports, and the original amateur goals of it, was largely a byproduct of the upper- and upper-middle-classes having sufficient leisure time to do so, while the working-class was unable to pursue such endeavors). This went doubly for international competitions like the Olympics, which the Soviet Union took no interest in until after the Second World War (instead hosting a socialist-themed Spartakiad in the 1920s and 30s).
This changed after the Second World War though, and efforts were made to develop Soviet sports as a means to showcase the superiority of socialism against the capitalist world, and that they could produce better athletes. With that the first Soviet Championship began play in December 1946, and the Soviet national team played at their first World Championship in 1954, dominating the international stage until its dissolution in 1991.
However as noted, Soviet hockey developed on it's own path. As noted, most European states were introduced to hockey by Canadian expats, and grew the sport that way. This was not the case in the Soviet Union, not exactly at least. Instead they adapted players from the sport of bandy, and trained them in hockey. For those unaware, bandy is sort of a mix of hockey and soccer: played on a large sheet of ice with 11 players per side, and shorter sticks than ice hockey uses. It was, and is, very popular in the Russian world and neighbouring countries, like Finland and Sweden. It was also crucial for development in Soviet hockey: to the size of a bandy rink measures 90-110 m (300-360 ft) long and 45-65 m (150-215 ft) wide, which forces players to be strong skaters and emphasizes passing plays and less physicality. Contrast this to the size of a hockey rink: NHL regulations are 61 m (200 ft) long and 26 m (85 ft) wide, while international is 60m (195 ft) by 30m (98 ft).
With players coming directly from bandy, and soccer (most Soviet players competed in both sports at the time), it was logical they would focus on what they were familiar with: skating and passing plays, and not physicality. Fighting was not really a part of the game, and so never really developed, and as the Soviet Union followed the international standard, there was never any need to focus on that later on, even when they started to play teams from North America, and the NHL in particular.
There has recently been some worthwhile books that look at hockey in a global context, which is great as the sport is indeed far more than the NHL, and if you want to read more about the game they are worth seeking out:
The Red Machine: The Soviet Quest to Dominate Canada's Game by Lawrence Martin (1990). Though it's over 30 years old now, this is still the standard book on the history of Soviet hockey. Martin, a Canadian who lived and worked in Moscow as a journalist throughout the 1980s, so was very familiar with the sport.
Hockey: A Global History by Stephen Hardy and Andrew C. Holman (2018). Written by academics, as the title suggests it's a global history of the sport, and brings context to the development of the game in Europe and Russia. It is also a very accessible read.
The Fastest Game in the World: Hockey and the Globalization of Sports by Bruce Berglund (2020). Similar to the title above, this looks at how hockey grew from a regional sport in Canada to the global game it is now. Berglund takes particular focus on some European countries and how the sport grew there, including Russia/the Soviet Union, so it is also highly recommended.