What is the significance of the term "separable" in the context of countability properties?

It's not a very good term and you're right that it has nothing to do with separation axioms. The intuition comes from thinking about, for example, $\mathbb{R}$: a subset $S \subseteq \mathbb{R}$ is dense iff any two distinct real numbers $a < b$ can be separated by an element of $S$ in the sense that there exists $s \in S$ such that $a < s < b$.

Second countable has an old-fashioned alternative name "completely separable" (or "perfectly separable"), and indeed second countable is a stronger property than separable.. (just as completely normal is stronger than normal, and completely regular stronger than regular..)

I've always found second/first countable somewhat nondescript names too. Nowadays we can just state the conditions as $d(X)=\aleph_0$ (separable), $\chi(X)=\aleph_0$ (first countable) and $w(X)=\aleph_0$ (second countable); the historical names can be a bit confusing (the same might be said of "normal" and "(completely) regular" spaces).

The handbook of history of general topology might have more info on its first use (and who came up with it). Engelking (General Topology in his historical notes) says (and I believe him) that Fréchet first defined the notion in (his doctoral thesis)

Fréchet, M.M. Sur quelques points du calcul fonctionnel. Rend. Circ. Matem. Palermo 22, 1–72 (1906). https://doi.org/10.1007/BF03018603

(but I don't have access to a digital library to view it). So it's an old name, probably one of the oldest still in use in topology (next to open/closed, limit point..)