I agree the fact that the character has wildshape has no bearing (no pun intended) (well maybe a little) on the Changelings shapechanger feature.
However I don't think the shapechange ability of a Changeling is that cut and dry. The Changeling specifically says it polymorph into any humanoid it has seen. Polymorph says you take on all the statistics of the creature you change into (including mental stats). Given the Doppelgangers history of taking on the forms of the specific people for infiltration. it seems to me the intent is that the Changeling takes on all the stats of the specific individual they're trying to mimic, while retaining their own alignment and personality. There's nothing in the Changeling description that overrides any of the wording in polymorph accept that creature being mimicked is a humanoid instead of a beast.
I'm pretty new to 5e though so there may be some other rules about polymorph that I haven't seen.