Let $H$ be the left half plane $\{z \in \mathbb{C} |\ Re(z) < 0\} $ $D^* =\{ z \in \mathbb{C} |\ 0<|z| <1\} $ then $exp : H \to D^* $ is the universal covering of $D^*$.
So the surface in the picture is diffeomorphic to $H$? I think it should not be. I am confused what is the relationship between covering space and graph of the multi-valued inverse of the covering map.
The picture on the left can be parametrized as the helix $\mathscr H$:
$$ (r, t) \mapsto (r\cos t, r\sin t, t).$$
where $r\in (0,1)$ and $t\in \mathbb R$. Note that the above mapping is a diffeomorphism from $(0,1)\times \mathbb R$ to the helix, and $(0,1)\times \mathbb R$ is diffeomorphic to $H$.
Thinking of the helix as the universal cover of $D^*$ has an obvious advantage: The covering map $\pi: \mathscr H\to D^*$ can be written as
$$ \pi(r\cos t, r\sin t, t) = (r\cos t, r\sin t).$$