I know it sounds crazy to call Siberia densely populated but compared to other places with similar climates like northern Canada and Greenland it is actually very well populated. There are many major cities like Novosibirsk, Omsk, Krasnoyarsk, Irkutsk, Barnaul, Kemerovo, Tyumen, Tomsk, Novokuznetsk, Khabarovsk, Vladivostok and arguably Chelyabinsk and Yekaterinburg, all of which have populations over 500,000 (Novosibirsk, Omsk, Krasnoyarsk, Chelyabinsk and Yekaterinburg each have over 1,000,000), and so many more with 100,000+. Most of these places have very long and brutal winters, comparable to Whitehorse or Winnipeg, Canada's only city with similar population and climate to the Siberian cities, and colder than Nuuk. (except Vladivostok which is comparable to wonderfully warm Duluth,_Minnesota) There are even cities with populations in the hundreds of thousands (ie. Yakutsk, Norilsk) that have some truly insane winters that are colder than any other cities in the world and comparable to and often even colder than Arctic research stations. In comparison. northern Canada's largest city, Whitehorse, only has a population of around 25,000 and the Yukon, Northwest and Nunavut territories together only add up to just over 100,000 people. Greenland only has ~56,000. Why does Siberia have so many large cities in such inhospitable climates?
First we should situate ourselves a bit as to where the populated parts of Siberia are. I posted this yesterday, but again for reference here is a population density map of Russia. The populated parts of Sibera (which for the purposes of this discussion we'll use as shorthand for "Asiatic Russia east of the Urals") are very much along its southern border, being wider towards the Urals and narrowing out and eventually stopping around the Kutnezsk Basin (Kuzbass). There are scattered areas of higher population around cities along the southern border to the Pacific, notably Irkutsk, Khabarovsk and Vladivostok.
As for why those areas are developed, I got a little into the background of that in an older answer I wrote that ironically was looking at things from the opposite perspective (ie, why is Siberia so underdeveloped). The populated areas of western Siberia are actually very rich grain-producing regions, which help support higher populations, especially in the cities, and many of those cities were developed in the 19th and 20th centuries as big industrial centers due to their location near major iron and coal deposits.
In many ways, the comparison to the development of Canada is helpful because in broad brushstrokes there are certainly parallels. Both Siberia and what is now Canada were largely explored and claimed by Europeans in order to control the fur trade: although Russians made it to the Pacific by the 17th century, much of the control of the region was based off of forts and trading posts designed to control native Siberian populations and monopolize the lucrative trapping and export of furs, especially sable. It was really in the 19th and 20th centuries that agricultural settlers from European Russia and Ukraine were encouraged to settle in Siberia, and starting with the First Five Year Plan of 1928-1932 the Soviets heavily pushed the development of new industrial centers in the region as well, often creating new cities from scratch, such as Magnitogorsk, and more infamously Norilsk. Much of this settlement in Siberia was both a spur to build a transcontinental railroad in the Trans-Siberian Railway (started in 1891 and completed in 1916), and that railway in turn determined where much of the subsequent population and development would be. This is a bit similar to the history of the Canadian Pacific Railway on the development of Canada. In any event, I would say that the more populated parts of Siberia if anything resemble the climate and history of the Canadian Prairie Provinces than they would, say, Yukon or Northwest Territories.
In any event, even with those similarities and narrowing our comparisons, we'd still be left seeing, say, Winnipeg with 700,000 or so residents, and Novosibirsk with almost a million more people (1,600,000). So what gives there?
Part of it is again just the nature of geography, and how it impacted climate. Take a look at the Koeppen map for Asia. You'll note that the temperate continental climate/humid continental climate zone in Siberia is pretty narrow, and also correlates very closely to the densely-settled parts of Siberia (as a bonus, I'll also point out that the areas of that zone in Kazakhstan were also heavily settled by Russian colonizers and are very ethnically Russian even today). North of that zone, and you get much harsher climate, plus permafrost, that discourages settlement. South of that zone, you get arid steppe and eventually desert. Pretty much if you're an expanding Russia, that area is the sweet spot for settlement and development.
Now let's look at the Koeppen map for North America. You can see that same area of climate, but while it covers the Canadian Prairie Provinces, it also covers much of the US Midwest, and even the Northeast. And just to the south of it are climate regions even more favorable to large scale agriculture. The big barrier doesn't so much go west-to-east as north-to-south in the form of the Great Plains, the Rocky Mountains, and the Basins and desert systems. So while Canada did see much of the same sort of settlement and development as Siberia, and at the same time, it also was massively offset by vastly bigger opportunities for European/white settlement to its south. Siberia didn't really have this. It would be like if instead of the Eastern United States, Canada was bordered to its south by steppe and desert.
ETA - here's a population density map for North America to show how the eastern half of the continent, but especially the Eastern United States, took much of the European / white settlement of the continent.
