Stochastic processes in continuous time are complicated, because they in a sense simultaneously live in two spaces separately.
More abstractly, a stochastic process is just a random variable $X$ taking values in a function space, for example the space of continuous functions with input $[0,T]$ taking values in $\mathbb R$, we typically denote this space as $C([0,T])$ and call it Wiener space. As a concrete example, the function $t \mapsto t^2$ on $[0,T]$ is a member of this space, but obviously not very random. So in this sense of the definition, for each $\omega \in \Omega$ we actually get a full trajectory, not just a value at a particular timepoint - the random variable is defined for all timepoints simultaneously.
When we think about a stochastic process at a given timepoint, we are actually thinking about the projection from the random variable down to the space it is taking values in. Concretely, we define $t \mapsto X_t(\omega)$ as a map taking values in $\mathbb R$. This is also a random variable with some associated probability distribution. One example is Brownian motion, which at every $t$ has a normal distribution with mean 0 and variance $t$, so $X_t \sim \mathcal N(0, t)$.
There is much more too it (including proving that such processes with such-and-such features even exists), but hopefully the distinction between the entire process as a random variable and its behaviour at any particular timepoint is made clearer.