As far as I know, no 18/19th century empire tried to colonize Siberia or the northern Far East besides Russia.
That sounds implausible to me and I'm curious how could that be?
Siberia was colonized earlier than the 18/19th centuries. There actually were a few challenges by great powers to Russia's colonial empire as it expanded and later: from China to some extent early on, if one is willing to count the latter as a great power in the 16-18th centuries, and later from Great Britain and Japan in the 19th century.
Russia conquered Siberia during a period when the western great powers were busy colonizing more accessible areas of the world, such as America, the Spice Islands, and India - i.e. places with trade goods that were arguably more desirable than fur (and even North America had its fair share of the latter).
China was Russia's most direct - and only, really - challenger in the area until the 19th century, resulting in a number of territorial wars as Russia progressed eastwards.
By the time Russia had settled its way to the Pacific, challenging its hegemony wasn't so much about the Siberian region itself (which is landlocked for all practical purposes) as it was about containment and settling durable borders.
British rivalry with Russia reached its climax in the 19th century, particularly in Central Asia. Russia wanted warm water ports in the India Ocean. Great Britain feared that this might open an invasion route to India. Russia further feared British commercial and military incursions in Central Asia. The whole region was an arena of tension then. The two fought a number of land grab related wars in the area - though not between themselves - until borders settled in the late 19th century.
This rivalry had consequences on the far eastern front, too. Russia sold Alaska to the US in the aftermath of the Crimean War. The Russian fear at the time was that they'd lose the territory to the British without compensation should another war break out between the two. Better Alaska in US hands, the thinking went, because the US wasn't much of a British friend then.
The eastern frontier then became the main theater of tensions: rivalry with Japan over the two empires' ambitions in Manchuria and Korea started at around the same time as things calmed down in Central Asia. The two empires' relationships went from mostly friendly to hostile when Russia opted to support China in the aftermath of the first Sino-Japanese war. The Russo-Japanese War erupted a decade later, ultimately forcing Russia to abandon its expansionist policies there.
There are basically two reasons: climate, and accessibility.
The first reason relates to the fact that Siberia (northern Asia) is the coldest part of the inhabited world. The only other part of the world that is "almost" as cold is European Russia. Basically, the Russians were the only people that could "stand" to live in most of northern Asia.
The second factor is "accessibility." Russia is adjacent to northern Asia from the west. China is adjacent to northern Asia to the south. There was a third "country," Mongolia, that actually started in north Asia. And here's where the climate factor comes into play: The Chinese weren't nearly as eager to colonize North Asia, even though it's much "closer" (on longitude) because it's much colder. The Mongolians started there and left for warmer climes in the south: China, Persia, the Ukraine. (Today's Mongolia is southeast of the original homeland around Lake Baikal.) It was the Russians, on the same "latitude" as north Asia, that found it worth colonizing.