Given the likelihood $p(x;\theta)$, we want to construct an estimator $\hat\theta(X)$ that takes in the observation $x$ and returns an estimate of $\theta$. There are many different ways to evaluate our estimator $\hat\theta(X)$. For example, the bias of $\hat\theta(X)$ is $$ \mathbb{E}_{p(x)}[\hat\theta(X) - \theta] $$ and the mean squared-error (MSE) of $\hat\theta(X)$ is $$ \mathbb{E}_{p(x)}[(\hat\theta(X) - \theta)^2] $$ Both the bias and the MSE are functions of $\hat\theta$ and $\theta$, and it seems at first glance that they are examples of loss functions, since loss functions are also functions of $\hat\theta$ and $\theta$.
More generally, in the context of statistics, does this mean that loss functions are only used to evaluate estimators?