Despite what some comments have claimed, such expressions are common in Bayesian probability and statistics and do have meaning in terms of conditional densities with respect to parameters and hyperparameters.
For instance, we could define a hierarchical model in which $$X \mid \mu, \sigma \sim \operatorname{Normal}(\mu, \sigma^2),$$ and $\mu$ is itself a random variable: $$\mu \sim \operatorname{Normal}(\mu_0, \sigma_0^2).$$ In this model, $\sigma$ is fixed and known; $\mu_0$ and $\sigma_0$ are hyperparameters representing the prior distribution of $\mu$. Then based on an observed sample $\boldsymbol X = (X_1, X_2, \ldots, X_n)$, we can compute a posterior density for $\mu$ given the data $\boldsymbol X$. We write this as $$f(\mu \mid \boldsymbol X, \sigma, \mu_0, \sigma_0) = \frac{f(\boldsymbol X \mid \mu, \sigma)f(\mu \mid \mu_0, \sigma_0)}{f(\boldsymbol X)},$$ which is essentially Bayes' theorem. Here, $f(\mu)$ is the prior density, and the denominator is the marginal density of the joint distribution of the sample $\boldsymbol X$.
The takeaway is that the symbols that are written after the $|$ sign are intended to represent those quantities that are fixed with respect to those symbols written before the $|$ sign. Note that if we wanted to, we could write $$f(\boldsymbol X \mid \mu, \sigma) = f(X_1, X_2, \ldots, X_n \mid \mu, \sigma),$$ showing explicitly that the joint density is a mapping $f : \mathbb R^n \to \mathbb R$.