No, druids are not considered "shapechangers".

Every creature that has the Shapechanger tag has a non-magical polymorphing ability: The [Shapechanger creature] can use its action to polymorph into a...

What counts as a Shapechanger 5e?

A creature must be designated — with a tag or other text — as a shapechanger to count as one. A creature might not have this particular tag although be a "shapeshifter" or "can take any shape", which might give a temptation to consider it a shapechanger in terms of Polymorph.

Can a druid be Polymorphed 5e?

Unfortunately, druids don't have access to True Polymorph like bards, warlocks, and wizards. Despite all that, both spells have no effect on shapechangers so wouldn't work on druids. "Shapechanger" is a monster tag, and doesn't apply to PC druids.