I've noticed that on the beast stat sidebar when added to my druid has a lower passive perception than my regular druid does. This druid has the Observant feat, and it doesn't seem to carry over to the Wildshape. Is this an oversight or how it is supposed to be?

The sidebar is merely for quick reference and having it track things like feats is difficult. The sidebar for creatures isn't really designed to be truly distinct from wildshapes, familiars, companions and such as. Given how most "creature" things do not carry your feats over the sidebar doesn't really factor them.

The sidebar is used as a "quick reference" not "stat replacement". If you are using it for wildshapes and you have feats altering stats in ways normally beyond the limits of "creature stats" then make a note somewhere else. You can add notes by customsing it.  It's still more convenient than pen and paper.

Since this was not intended to be comprehensive and only "quck reference" and not really a druid specific thing this is neither a bug or an oversight. Perhaps they'll create wildshapes as a separate feature in future which could allow them to have it be more relevant to wildshape rules (which are different to pretty much everything else and use creature stats differently than most other things) but right now it isn't this and therefore doesn't factor feats very well.