Every creature that has the Shapechanger tag has a non-magical polymorphing ability:
The [Shapechanger creature] can use its action to polymorph into a...
while every creature that seems to have an inherent shapeshifting ability but does not have the Shapechanger tag has a polymorphing ability which is described as 'magically polymorphing':
The [Creature] magically polymorphs into a...
It seems that if the shapechanging ability is magical in nature (as Wild Shaping is described as being), it is not considered part of the creature inherently (and thus not part of its creature classification, in terms of the Shapechanger tag), but rather as an 'addition' or learned behaviour.
As such, while under a magic-suppressing effect (such as being in an Antimagic Field), a creature with the Shapechanger tag is still able to shapeshift, while a creature without it is not, thus demonstrating it is not an ability inherent to the creature and the creature (Druid included) is not considered a shapechanger.