It's good that you pointed this out and it would be helpful if the specific documentation errors were described. For example the documentation for Mean with weighted data shows

The $\frac{1}{n}$ should not be in that definition.

I also think there is a problem with WeightedData and the documentation... With the same data generation:

Mean@d
Mean[WeightedData[d, 
  w]] (*Correct, but different from the documentation of Mean!*)
Mean[d w]/Total[w] (*From the documentation of Mean for WeightedData *)

As fo the variance, the problem seems linked with unbiasedness :

Variance@d  (*unbiased*)
Mean[(d - Mean@d)^2] (nbpoints/(nbpoints - 1))(*unbiased*)
Variance[
 WeightedData[d, 
  w]]  (*  Problematic definition in the documentation, \
for  WeightedData*)
Mean[(d - Mean@d)^2]  (* biased *)

Does this make sense? I don't think so.

The rule is: equal weights = without weights.