You definitely would run into identification issues. Suppose $f(x|\theta) = x \theta$. Then multiplying $\theta$ and $\sigma$ by any common factor would result in an identical situation. In this case the MLE would not really be defined.

Generally, as $\theta$ changes, $X$ is revealing a different facet of $V$. For example, if $f$ is a projection map and $\theta$ determines the angle of projection, then $f(V|\theta)$ is the shadow of $V$ in the direction of $\theta$. Many of the shadows could be normally distributed, and could be essentially unrelated to each other.

You definitely would run into identification issues. Suppose $f(x|\mid\theta) = x \theta$. Then multiplying $\theta$ and $\sigma$ by any common factor would result in an identical situation. In this case the MLE would not really be defined.

Generally, as $\theta$ changes, $X$ is revealing a different facet of $V$. For example, if $f$ is a projection map and $\theta$ determines the angle of projection, then $f(V\mid\theta)$ is the shadow of $V$ in the direction of $\theta$. Many of the shadows could be normally distributed, and could be essentially unrelated to each other.