I have a book that says otherwise. It's called America Goes To War and
it has a list of the members of the Axis. Finland's on that list.
The "Allies/Co-belligerents" discussion is a tricky thing to get into. To be honest, there is no consensus even in Finland, which is the correct reading of the situation.
For alliance:
- pre-meditated, concerted plans for the Finno-Soviet Front, which could be seen as a minor theatre of the Eastern Front
- defensive responsibility of Lapland handed over to the Wehrmacht to allow Finnish troops to be stationed in Karelia
- Finnish dependence on German war material and supplies
For co-belligerence
- no formal device of alliance was signed (until the "Ryti-Ribbentrop Pact" of 1944, which in itself was a ruse to get Finland out of the war), thus Finland wa not a part of "Axis Proper"
- as Kriegdämmerung pointed out, Finland only fought the Soviets and made this plain to the Western Allies
- Finland had clearly distinct war aims from Germany and the Finnish High Command regularly declined to act on German requests, for example concerning the Murmansk railroad
- no regular Finnish troops took part of Barbarossa under German command
The co-belligerence theory is one invented and propagated by the Finnish govenrment and the High Command in the event. And it did catch on quite well indeed, as the whole point of the operation was to save Finland from excessive Allied wrath and retaliation if Germany manages to lose. A way of hedging the bets, as it were, like many other actions of Finnish wartime leaders who were damn lucky in their efforts.
If I had to say something, I'd contend that Finland was a German ally, but not a member of the Axis, if someone can see any sense in that. Finland was always in the war only for itself, to regain lost territories, to create a defensible border. It joined the war when it seemed opportune and abandoned Germany when the going got too hot, maintaining a certain freedom of action in the between. I'd say this relative freedom of action is something that distinguishes Finland from most, if not all of Germany's "Real Axis " allies.