Why is $D_4/Z(D_4)$ "just like" $V$, Klein's 4-group? What makes the two similar? $D_4/Z(D_4)$ here is the quotient group of $D_4$ by its center.

There are a few computational details missing, but I will leave those for you to do.  After some computation, you will find that
$$ Z(D_4) = \{ e, r_2 \}, $$
where $r_2$ is the rotation of the square by $\frac{\pi}{2}$ radians.  Now $|Z(D_4)| = 2$, and so (recall the center of a group is always normal)
$$ |D_4 / Z(D_4)| = 4. $$
At this point, we may say that $D_4 / Z(D_4) \cong \mathbb{Z} / 4 \mathbb{Z}$ or $D_4 / Z(D_4) \cong V_4$ (the Klein $4$ group) because these are the only two groups of order $4$ up to isomorphism. Check that each element of $D_4 / Z(D_4)$ has order $2$, which implies $D_4 / Z(D_4) \cong V_4$.
