In section 1.4.5 of Kevin Murphy ML textbook, he introduces linear regression where for a given data $x$, the target $y$ is assumed to be obtained through
$$y(x) = w^Tx + \epsilon, \text{ where } \epsilon \sim \mathcal{N}(\mu, \sigma^2)$$
Since $\epsilon$ is a random variable, therefore $y$ is a random variable as well induced solely by $\epsilon$.
However, the author then defines $$p(y|x, \theta) = \mathcal{N}(y|\mu(x), \sigma^2(x))$$
where $\theta = (w, \sigma^2)$
First of all, $x$ here is explicitly just a vector with no assumption that is was generated according to a distribution. Secondly, $\theta$ here is just the parameters associated with the Gaussian as well as the model, there is no probablistic interpretation as well.
How does the author condition on two deterministic variables?
Relevant text below: