These attunement rules can be summarized, 'if a magic item requires attunement, all of its magical features require attunement, unless a particular feature says it does not require attunement.' Without attuning to the berserker axe, it functions as a mundane axe, since none of its magical features state that they function while not attuned.
When a magical feature of a magic item does not require attunement, the attunement rules tell us that the feature will explicitly tell you that it doesn't require attunement. Without becoming attuned to an item that requires attunement, a creature gains only its nonmagical benefits, unless its description states otherwise. The rules for attunement are unambiguous:
When a magic item requires attunement, you must attune to get the magical benefits.
