Yes, the two shapes must be similar. If you draw segments $AC$ and $EG$, then you have two pairs of similar triangles. $\angle A \approx \angle E$ and $\angle D \approx \angle H$ so $\Delta ACD \sim \Delta EGH.$ Similarly $\Delta ACB \sim \Delta EGF.$ Because their two constituent triangles are similar, we must have $ABCD \sim EFGH.$
No, it is possible the shapes are not similar. Consider the example above of two rectangles. All their angles are congruent, but the shapes are clearly not similar.
I had just introduced the class to the AA Similarity Postulate - the idea that if triangles have two pairs of congruent angles, then one is a scaled-up version of the other, i.e. they are similar. I posed my class the following question.
Is there a corresponding AAA Similarity Theorem for quadrilaterals?
I was impressed by how clearly my students argued two different views on the question.
Who do you believe?
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Hanoi, Vietnam. Your comments are always welcome.