I have found myself going down a rabbit hole on this and wondering if anyone can clarify.

Is the Wildshape ability of the Druid magical?

The specific rule text makes no mention of it being a spell, a magic action (as it is now a bonus action) and no text describing it as 'magical'.

However, previous editions described it as magical and the description text of the class describes it as "Harnessing the magic of animals, plants, and the four elements, Druids heal, transform into animals, and wield elemental destruction."

This one has me baffled. 

Yeah, this is a weird one. As written it doesn't meet any of the criteria in the definition of "Magical Effect" in the Rules Glossary:

An effect is magical if it is created by a spell, a magic item, or a phenomenon that a rule labels as magical.

But it seems pretty clear that the designers generally consider it to be magical; for instance, it's referred to as a magical effect in this Sage Advice question:

Can you use Dispel Magic to dispel a magical effect like a Druid’s Wild Shape?

Dispel Magic has a particular purpose: to break other spells. It has no effect on a magical effect that isn’t created by a spell unless the text says otherwise (though the DM can always make exceptions).

I generally think of it as a magical effect, and I would always rule it that way when I'm DMing, but I think by a very strict reading of the rules, it is not magical, because it doesn't meet those criteria in the Rules Glossary.