Affirming The Consequent?

This is not if A then B. B. This is actually a valid form of an argument:

Only X are Y.
Z is Y.
Therefore, Z is X.

The difference is "only" vs. "if/then". If you think about it, it makes sense. If we say that ONLY members of group X can be members of group Y, and Z is a member of group Y, then by necessity, Z must also be a member of group X. Here is another example:

Only grandmothers can give fruitcakes as gifts for Christmas. (i.e., if someone gives a fruitcake as a Christmas gift, that person MUST be from a grandmother.)
Ida gave a fruitcake as a gift for Christmas.
Therefore, Ida MUST be a grandmother.

This is valid in that if both premises are TRUE, the conclusion MUST be true.

It is an argument that seems logically sound but really isn't.  It is a form of circular reasoning because we have no objective definition of Targaryen to begin with.  What is a Targaryen, at maximum it is all the descendants of Aenar Targaryen, but he lived a long time ago which means a lot of people are Targaryens based on that definition. 

You cannot really make logical inferences based upon a premise which is arbitrary.  Here we are trying to prove that someone belongs to an arbitrary group but without defining the group, all we are doing is defining the first premise with it's conclusion (circular reasoning).