As Andre said, categoricity means that all models are isomorphic and $\kappa$-categoricity means that all models of size $\kappa$ are isomorphic.
The problem is that model theory is usually developed within a universe of ZFC. Things within think of that universe as a class, not as a set. If $\frak M$ is a countable model of ZFC living as a set inside a larger universe $V$, the model theory inside $\frak M$ does not have to coincide with the model theory inside $V$.
In fact this is true for class models, that is $V$ can be a "very large" model of ZFC, and there is a subclass of $V$ which is also a model of ZFC that is significantly "smaller" and these two models have different model theories (because they ultimately contain different sets).
When we say that a model of ZFC is countable, we come from an external point of view. That is we live in a very large universe and that universe happened to know a countable set $M$ with a binary relation $E$ such that $\mathfrak M=\langle M,E\rangle$ is a model of ZFC.
It is possible that $M$ has a subset $N$ such that $\mathfrak N=\langle N,E\cap(N\times N)\rangle$ is also a model of ZFC, and it is possible that these models are not isomorphic at all.
This tells you that ZFC is not $\aleph_0$-categorical, and that categoricity here is far from categoricity in philosophy.
Furthermore, Cohen's work on forcing shows us that countability is far from absolute. If $\frak M$ is a countable model of ZFC, then there is a "slightly larger" $\frak A$ which is a countable model of ZFC, but $\frak A$ thinks that $\omega_1^\frak M$ is countable. Namely we took a set that $\frak M$ thought is uncountable, and we added bijections between this set and $\omega$.
However, as I said before we do all these model theoretic considerations inside a fixed universe of ZFC (this universe is not a set in our context), and the question whether or not a set is countable is answered in this universe. The notion of countability, therefore, does not "go down" to models which live inside this universe. They have their own notion of countability.