Finland was categorised as an ally of Germany, and an enemy combatant. This is, IMO, a quite accurate description. Despite Finland presenting herself as a co-belligerent with Germany rather than an ally of Germany, "ally" is accurate. While there was no formal treaty of alliance, Finland's entry into the war after Barbarossa was planned in advance, with Germany. If "enemy combatant fighting as an ally of Germany" is what you mean by "Axis power", then there is no mystery in the "why".

If, however, you mean that Finland was categorised as a member of the Axis, this wasn't the case.

In the treaty itself, the relevant text is:

Whereas Finland, having become an ally of Hitlerite Germany and having participated on her side in the war against the Union of Soviet Socialist Republics, the United Kingdom and other United Nations, bears her share of responsibility for this war.

Bulgaria, Hungary, and Romania are described similarly in their treaties (adding the USA to the USSR and UK as an enemy during the war).

Italy was described as a "party to the Tripartite Pact with Germany and Japan".

Thus, of the five enemy combatants in the Paris Peace Treaties, Italy is described as a member of the Tripartite Pact, and the others as allies of Germany.