My understanding of the main difference between Frequentist vs Bayesian stats is that the former treats parameters as variables with fixed but unknown values whereas the latter treats them as random variables.
From that perspective, I get the impression that priors (and posteriors) would only make sense in Bayesian statistics. Is this understanding correct?
You are right in the sense that in Bayesian statistics the space of parameters $\Theta$ is endowed with the structure of a probability space $(\Theta, \mathcal{A},\pi)$ where $\mathcal{A}$ is a $\sigma$-algebra and $\pi$ a probability (prior) so it make sense to ask questions of the form $$ B\in \mathcal{A},\quad \int_B f(\theta)\pi(d\theta) $$
Where this would not make sense in a frequentist setting.
