Question

Isosceles triangles are always similar according to the Angle-Angle Similarity Postulate?

Answer 1

I don't think so. 

The Angle-Angle Similarity postulate says that two triangles are similar
if they have two corresponding angles that are equal in measure ... two
angles in one triangle equal to two angles in the OTHER triangle.

Every isosceles triangle has two angles that are equal to each other. 
But that doesn't tell you how those angles compare to the angles of a
DIFFERENT isosceles triangle. 

If you pick two isosceles triangles, there's practically a zero chance
that the two equal angles in one triangle have the same measure as
the two equal angles in the other triangle.  So the Angle-Angle Similarity
Postulate doesn't apply to them.