It would appear that when you go a multiclass of moon druid and monk not all the forms lets you apply unarmored defense like it should.

Some do, but ones like say owlbear do not, but the giant spider and panther forms will. Some forms also allow bonus to AC from dual wielder, while others will not. Dual Wielder feat is one of the few feats you can take that actually help you in forms to boost AC in forms. Also unarmored defense is one of the few class features that work in forms too, technically fighting style defense also work... provided your wearing armor before you wild shape