Say we have sample from a population that follows an AR(1) process: \begin{equation} x_t=\rho x_{t-1}+\varepsilon_t \end{equation}
Is it correct to assume that $x_0$ is a constant? Or does the $x_0$ that is often regarded as the initial value refers to the population $x_0$?
You can always specify $x_0$ to be drawn from a given distribution (a constant is a special case). In that case, a solution is just given by iterating forward according to the model.
However, if you want a covariance stationary solution $\{ x_t \}$, then $x_0$ (or any other $x_t$ for that matter) necessarily cannot be deterministic.
When $\{ \epsilon_t \}$ is i.i.d., then covariance stationary solutions are also strictly stationary and all $x_t$'s have the same distribution, including $x_0$.
Usually, $x_0$ is assumed to be a realization of a random variable. The realization will differ across different sample paths (time series) from the same data generating process. But it is also possible to specify a model where $x_0$ is fixed.
