It is a really good question. In fact, in the "standard" settings $\theta$ runs $(-\infty,\infty)$ but $\Theta$ must be $2\pi$-periodic. So problem is

\begin{align}

&\Delta u=0,\label{eq1}\\

&u|_{r=a}=g(\theta),\label{eq2}\\

&u(2\pi)=u(0), \ u_\theta(2\pi)=u_\theta(0).\label{eq3}

\end{align}

On the other hand, we can consider membrane also in the same shape, but it is "clumped" along $\{\theta=0\}$, in which case $\theta$ runs $(0,2\pi)$ and (\ref{eq3}) is replaced by

\begin{align}

&u(2\pi)=u(0)=0.\label{eq4}

\end{align}

We can also consider membrane also in the same shape, but it is "cut" along $\{\theta=0\}$ and both sides of cut are left free in which case $\theta$ runs $(0,2\pi)$ and (\ref{eq3}) is replaced by

\begin{align}

&u_\theta(2\pi)=u_\theta (0)=0.\label{eq5}

\end{align}

And so on. These are *different* problems. Separation of variables leads to different decompositions.