In summary, the Winter War and the Continuation War were regarded as separate from the rest of World War II because Finland and the Soviet Union, and many of the other potential participants were happy with it that way. Germany would have benefited from the Continuation War being more closely tied to its own war with the Soviet Union, but Finland was willing to put its own desires ahead of Germany's in this regard.

First, the Winter War. Neither Finland nor the Soviet Union wanted the war to be considered part of the greater war. Of the main countries already involved in World War II, Britain and France wanted to avoid war with the Soviet Union, having already avoided declaring war on the Soviet Union when it invaded eastern Poland. Poland, occupied, had little choice about involvement in Finland's war. Perhaps Germany would have benefited greatly from Britain and France fighting the Soviet Union, but Germany did nothing more than block military aid to Finland (including aid from Italy). The Soviet Union wanted to avoid war with France and Britain.

Of potential participants, Norway and Sweden wished to stay neutral. France and Britain offered military aid, but their only feasible route to Finland was through Norway and Sweden. They would have been happy to bring Norway and Sweden into World War II on their side, or at least deny Swedish resources to Germany; these appear to have been goals in their plans to aid Finland. Finland would have been happy with substantial military aid, including troops, from Britain and France, but did not want a Franco-British occupation of Norway and Sweden with only token and ineffective forces reaching Finland. By the time there was any real possibility of troops from France and/or Britain, this was the most likely scenario. Finland also did not want any such intervention integrating the Winter War with World War II, since this would make Finland part of the Allies, and at war with Germany (and the Soviet Union). Given the distance of Finland from France and Britain, and its proximity to Germany, this would have been a recipe for disaster, if Germany joined in actively. Given the division of Europe into spheres of influence in the Molotov-Ribbentrop Pact, Germany might well have avoided direct involvement, but even a hands-off war with Germany at the time would have been bad for Finland.

Finland was quite willing to use the threat of Allied intervention to encourage the Soviet Union to offer acceptable peace terms, and the possibility of Allied intervention does appear to have encouraged such terms.

Second, for the Continuation War, Finland was very keen to keep its war separate from Germany's war. The name "Continuation War" was chosen by the Finnish government for propaganda purposes, and the official stance as a "co-belligerent" rather than an ally of Germany (despite involvement of both Germany and Finland in the planning for both Barbarossa and the Continuation War). Britain and the USA were both sympathetic to Finland, and were happy enough to accept this separateness. Both Britain and the USA urged Finland to withdraw behind the pre-Winter War borders (having already passed them by this time), and most definitely to not advance and cut the Murmansk railroad. Finland stopped its advance, and didn't cut the Murmansk railroad. Despite Finland not withdrawing to the pre-Winter War border, the USA didn't declare war on Finland. Britain, allied with the Soviet Union in the post-Barbarossa war, did declare war on Finland, but took no significant military action against Finland. Given the Finnish halt, including the lack of a Finnish attack on Leningrad, at a time when the Soviet Union was hard-pressed, the Soviet Union was happy enough with this state of affairs.

Germany would have benefited from more active and aggressive Finnish involvement, after Finland's halt. Germany wanted active Finnish participation in an attack on Leningrad; Finland refused.

Third, the Lapland War was clearly separate from World War II proper. Finland was fighting against Germany, but with the limited aim of expelling German forces from Finland (or disarming them). Finland's war finished officially before VE day, when the last German forces left Finland on the 27th April, 1945 (there had been very little fighting between Finland and Germany since early November 1944).

From a German perspective, the old narrative of 'muh defense against the Judaeo-Bolshevik hordes' has a Secret Protocols sized hole in it where the Winter War was the Germans selling Finland to the USSR until the Finns provided a much stiffer fight, with the Lapland War in turn marking that the German bid to wield Finland as their least murderous ally ultimately failed. Just like Putin's made it stick, not once, but twice against the Soviet Union. They made the USSR pay such a steep price the first time that the Karelian Soviet Republic that would have become the FSSR was negated after the 1939 campaign.

This is very much a thing of a historical sleight of hand encouraged even by the Russians, who have little to gain from the memory of the lengthy and bitter set of half-victories won by the poorly-led Soviet northern armies against the Finns, and by the Finns themselves, who have not the least inclination to accept that they were an Axis Power working with the genocidal barbarian hordes of Adolf Hitler in pursuit of an overall plan. To a degree, the Finns regaining everything taken from them in 1939 and parking their armies there rather than extending the campaign further at the expense of Leningrad helped with that. The Finns were more aware than the Germans in 1941 that the initial wave of victories did not mean the USSR was going to wither on the vine, and were preparing for the prospect of a postwar relationship with the USSR by doing as little as possible to enable the bigger war and simultaneously as much as possible to make them too stiff a nut to crack.

From a German perspective, the old narrative of 'muh defense against the Judaeo-Bolshevik hordes' has a Secret Protocols sized hole in it where the Winter War was the Germans selling Finland to the USSR until the Finns provided a much stiffer fight, with the Lapland War in turn marking that the German bid to wield Finland as their least murderous ally ultimately failed. Just like Putin's Russia, the Germans do not like to discuss at any great detail or length the period when the Germans and USSR were de facto allies and the USSR's bid to regain as much of the territory of the old Romanov Empire as they could received the full sanction and boost of Hitler's regime. The ability of Finland to defeat Voroshilov's offensive took the Germans by surprise as much as anyone else, but then they were hardly going to telegraph 'oh yeah, well....um...good job, Finland! We don't have to smile and accept the USSR annexing you, great performance!' the way it was so with the Baltic states and what are now the western territories of Belarus and Ukraine.

In the Russian case, where the USSR did not cover itself in glory the war was under-emphasized, it fits into what Glantz calls the 'forgotten war' which is about 30% of the war and some of its biggest and goriest battles. Suomussalmi and its like marked signal failures of Soviet arms that embarrassed the USSR globally and made it appear vulnerable enough for the Soviet Union's enemies to see it as a rotten structure a single kick would topple. The 1944 offensive did break through the Finns' southern lines and ultimately forced them out of the war but was less decisive than Bagration or the Balkan offensive, so it gets neglected in favor of them even in the 'Stalin's Ten Blows' old school Great Patriotic War narrative.

And in the northern theater, much of the Siege of Leningrad, where the bulk of the Continuation War was the northern sector of that Siege, fell squarely into a category of poorly managed battles that gutted Soviet forces and left the lines locked without much else changing. The Soviet Union had little interest in stirring those waters, so the Finns got to write and formulate the history of all of this to suit themselves even more than the Germans did, and neither Germany nor the USSR were particularly invested in challenging this, after the war.

Sources:

David Glantz, Stumbling Colossus

David Glantz, When Titans Clashed: How the Red Army Stopped Hitler

David Glantz, The Siege of Leningrad.