Assume you cast Barkskin in humanoid form:
You touch a willing creature. Until the spell ends, the target’s skin has a rough, bark-like appearance, and the target’s AC can’t be less than 16, regardless of what kind of arm or it is wearing.
It is clear that under the rules of Wild Shape "transforming doesn’t break your concentration on a spell you’ve already cast", but does the effect of the spell carry over to your new skin?
Unlike something such as Mage Armour, Barkskin doesn't set your base AC to a certain number, it just says that your AC can't be anything less than 16. Therefore you don't have multiple ways to calculate your AC and aren't forced to choose between one or the other (PHB pg.14).
Wild Shape also says nothing about ending any effects that are already applied to the target, so things like Conditions and spell effects would carry over from one form to the next: also discussed here regarding Exhaustion and Wild Shape.
I'll point out that Jeremy Crawford has also ruled that barkskin works in beast form.
Q: Can druids cast stoneshape or barkskin have that work when in wildshape?
A: Stoneskin and barkskin work in beast form.
