$\newcommand\th\theta$Given a family of pdf's $p_\th$ and a prior pdf $g$, the maximizer of 
$$F(q,g)(y):=E_q\ln p_\th(y)-D_{KL}(q\parallel g)$$
is always unique -- if any two pdf's differing only on a set of measure $0$ are considered to be the same. 

Indeed, letting $p(y):=\int d\th\,g(\th)p_\th(y)$ and $p(\th\mid y):=g(\th)p_\th(y)/p(y)$, we have 
\begin{align}
F(q,g)(y)&=\int d\th\,q(\th)\ln p_\th(y)-\int d\th\,q(\th)\ln\frac{q(\th)}{g(\th)} \\ 
&=\int d\th\,q(\th)\ln\frac{g(\th)p_\th(y)}{q(\th)} \\ 
&=\int d\th\,q(\th)\ln\frac{p(\th\mid y)p(y)}{q(\th)} \\ 
&=\ln p(y)+\int d\th\,q(\th)\ln\frac{p(\th\mid y)}{q(\th)} \\
&=\ln p(y)-D_{KL}(q\parallel p(\cdot\mid y)). 
\end{align}
So, any maximizer of $F(q,g)(y)$ in $q$ is a minimizer of $D_{KL}(q\parallel p(\cdot\mid y))$ in $q$ (and vice versa), and the [only minimizer of the latter Kullback–Leibler divergence][1] is the posterior pdf $p(\cdot\mid y)$. 


  [1]: https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence#Properties