Government statistics show that the real (adjusted for inflation)

In B it says that it fails to take into account inflation with respect of Anderson family income.

We know that inflation is measured for a basket of things with different weights.

What if Andersons bought 10 cars, whose weight in the basket is very low. But prices of cars he bought jumped a lot. Then purchases on Andersons cars will have high inflation compared to inflation for average basket of good. Then Andersons will have lower real income than if calculated with given inflation.

Why is B wrong then?

(B) is incorrect because it's not the case that the argument "fails to take into account inflation with respect to the Andersen family's income." The argument does take inflation into account, not in the way you suggested, but it does take it into account.

IMO
even if you take inflation into account i.e. even if you say that the flaw in the argument is that it did not take inflation in account, can you now conclusively say that real average income of Anderson's family has increased/decreased in the past ?
no still without knowing what the real average income was in the past for this family in question we cannot strengthen/weaken the argument.

whereas in D we can surely say that the real average income has decreased rather than increasing and thus weaken the argument and thus this is the flaw.

Government statistics show that the real (adjusted for inflation) average income for families has risen over the last five years. Therefore, since this year the Andersen family's income is average for families, the family's real income must have increased over the last five years.

The reasoning in the argument is most vulnerable to criticism on the grounds that the argument

(A)  ambiguously uses the term "average" in two different senses - WRONG. No, no such ambiguity arises, neither does it impacts.

(B)  fails to take into account inflation with respect to the Andersen family's income - WRONG. Inflation would not be specific to a particular family, it's common for all.

(C)  overlooks the possibility that most families' incomes are below average - WRONG. Most families may be below but we are only concerned about Anderson family's income.

(D)  fails to consider the possibility that the Andersen family's real income was above average in the recent past - CORRECT. POE helps. But if the income was above average in the past and now just average then real income has decreased in fact.

(E)  presumes, without providing justification, that the government makes no errors in gathering accurate estimates of family income - WRONG. Error or no error it does not matter as it would have been for all the families.

Answer D.