In a word, no.

The Winter War (1939-1940) was clearly Finland against the Soviet Union, and caused a major shock and re-evaluation of political views to many of those in the political right as Germany refused all entreaties for help against the "godless Communists." The official line that was generally accepted among the populace during the more controversial period, the Continuation War (1941-44) was that Finland was fighting a separate war against the Soviet Union, and just happened to have German (and other Axis, though these were rarely mentioned) "comrades in arms" doing the same. (The term "comrades in arms" was used even officially to refer to the situation.)

Finnish diplomacy put great efforts to trying to make the difference clear to the Western allies, who the overwhelming majority of the political elite and the populace preferred to win the Second World War. This succeeded in case of the United States, which never declared war and whose now unclassified State Department files show clearly that the U.S. government considered Finland to be separate from the rest of the Axis allies. (Also visible in the demands put to Finland vs. rest of the Axis, and accepted by Stalin.) In case of the United Kingdom, Stalin's insistence won and the UK, after correspondence between Churchill and Mannerheim, declared war on 6th December 1941. (Aside from one rather desultory air raid to Northern Finland, UK forces didn't actually participate however.)

Finnish High Command also generally tried to distance itself from the German demands and proposals for more integrated command structure and operations. The most visible results of this were that Finnish forces refused to cut off Leningrad from the north even when this would have been feasible, and when Mannerheim - very clearly thinking about the eventual post-war settlement - firmly vetoed a plan to cut the railway line to Murmansk harbor, a vital route for Lend-Lease shipments. Mannerheim, the old Czarist general, firmly believed that Russia would prevail somehow no matter what the Finns did, and that if Finnish forces cut off this important lifeline, there would be no leniency in the eventual settlement. (To be fair, at this point early in the Continuation War the now-attacking Finnish army had also suffered casualties equal to those suffered during the whole of the Winter War, and the suggested attack would have meant many, many more.)

However, already during the buildup to the Continuation War and for the duration of the war, the entire Northern Finland-Lapland front was in practice manned by German troops (augmented with Finns for local knowledge), and there certainly was significant cooperation between the two militaries. To a large extent, the cooperation became moot as the German mechanized troops, with equipment, training and mindset geared towards fighting in the Central European plains, proved totally unsuitable for operations in the North, and the planned German thrust towards Murmansk stalled very early on with practically no gain save for some square kilometers of frozen tundra to show of. For most of the Continuation War, the German troops in the North were basically stuck on a static warfare against equally passive Soviet adversaries.

German troops proved far more useful in Southern Finland in the epic fight of 1944, when the Finns - aided by a division of German troops and a Luftwaffe detachment, and very much helped by German handheld anti-tank weapons - stopped the Soviet general offensive. To receive this aid, German foreign office had extracted a promise from Finnish president Risto Ryti that he would never sign a separate peace with the Soviets. Giving the Germans a lesson in being very careful about the exact wording of the agreements they sign, Ryti promptly resigned after the aid was delivered and the Soviet assault had been stopped. Mannerheim, chosen by the Parliament to succeed Ryti, immediately signed an armistice with the Soviet Union.

The stipulations of this armistice demanded, among other things, the expulsion of German troops from Lapland. Initially, the Germans simply retreated to leave "pursuing" Finnish troops to occupy empty positions, but pressure from the Soviets led to Finns having to attack the erstwhile allies. The resulting Lapland War was more of a skirmish, but the Germans resorted to scorched earth tactics so effectively that practically all buildings in Lapland were destroyed, in addition to mines and booby traps being used widely. "Lapland burners" is still a term some in Finland use for Germans, and a common joke goes that one should be very careful if a German asks for a light or matches in Lapland.

These experiences, and the political and spiritual need to distance Finland from Nazi Germany, contributed to the popular belief that Finland was not a part of the Axis but had been compelled to join forces with the Devil as no other help was forthcoming. More recent research has challenged this claim somewhat, and it's a question of semantics (and of political preference) what specific term should be used for Finland's relationship with Nazi Germany. For example, there is solid evidence for Finnish High Command cooperating with the Germans in planning of the Operation Barbarossa - but only after a failure of Nordic defense union (due to Soviet pressure) and a serious war scare and Soviet pressure in 1940, pressure that most reputable historians believe had been aimed at eventual annexation of Finland. Likewise, there is clear evidence that the initial German successes against the Soviet Union created an euphoric feeling particularly among the political right, and many public figures (up to and including the future president of Finland, Urho Kekkonen) made statements they came to regret later. My personal opinion as a history buff is that the term "alliance" describes fairly accurately the military side of the things, but Finland's reasons for joining forces with the Germans (only after Nordic defense pact was torpedoed by the Soviets), overall war aims, domestic situation and wartime policy put her sufficiently separate from the rest of the Axis that the separate treatment the U.S. government insisted upon was quite justified. Then again, as a Finn, I may be biased!

There is little doubt however that the average Finn viewed both dictatorships - the one s/he was fighting against, and the one s/he was nominally allied with - with distrust and didn't think Finland was fighting for the Germans, or even that Finland and Germany had much in common besides a common enemy. Interestingly, the pro-German sympathies were likely stronger among the more educated segment of the populace, largely because they were also more likely to lean towards the right and because Finland had strong ties to the German university system - not to mention German "help" during the 1918 Civil War. Nevertheless, even there the Nazi regime was for the most part viewed with suspicion, if not despised outright, although some university professors were later even tried for their connections to the Nazi regime.

The above is based on a fairly broad reading of mostly Finnish language historical works, but I can point out sources to specific topics if so desired.