You probably mean to assume that $X$ is connected, and to allow only covering spaces $E\to X$ such that $E$ is connected.
The connected covering spaces of $X$ are the objects of a category, where a morphism from $E\to X$ to $E'\to X$ means a continuous map $E\to E'$ that respects the projection to $X$. In general such a map $E\to E'$ is not itself a covering space, although this is true in the good case, i.e. when $X$ is connected and locally path-connected and semi-locally simply connected.
In the good case, that category of connected covering spaces is equivalent to the category of sets with transitive action of $G=\pi_1(X,x)$. Note that the universal covering space is not initial; it corresponds to a set with a free transitive $G$-action; it has a map to every object, but not a unique one.
Certainly there are non-slsc examples in which the category of connected covering spaces has a "universal"object. For example, the "topologist's sine curve", which is connected but not path-connected, has no covering spaces except the one-sheeted one.