Here are two definitions that I have encountered:
The first, corresponding to this Wikipedia page, is the following.
Definition.$\ $ Let $(X,d)$ be a metric space. Let $\epsilon\in\mathbb{R}^{>0}$. An $\epsilon$-net for $X$ is a subset $S\subseteq X$ of points such that the collection of $\epsilon$-balls centered at those points forms a cover for X. i.e., $$S\subseteq X \ \text{ is an }\epsilon\text{-net}\ \ \iff\ \ \bigcup_{x\in S}B_X(x,\epsilon)=X\ \ .$$ or equivalently $$S\subseteq X \ \text{ is an }\epsilon\text{-net}\ \ \iff\ \ \forall x\in X,\ \ \operatorname{dist}(x,S)<\epsilon \ .$$
The second, corresponding to this Wikipedia page, is the following.
Definition.$\ $Let $(X,\tau)$ be a topological space. Let $(P,\preccurlyeq)$ be a partially ordered set with the additional property that $\forall a,b\in X,\ \ \exists c\in X\ :\ a\preccurlyeq c\ \land\ b\preccurlyeq c\ $. $\,$A net $x$ is a function $x\colon P\to X$ and is also denoted as $(x_\alpha)_{\alpha\in P}$.
How are these two concepts related precisely? Since all metric spaces are topological spaces, the names would clash unless every $\epsilon$-net is a net.
Is it true that every $\epsilon$-net is a net? Or is the similarity in the names a mere unfortunate coincidence? To me, it looks like the two definitions are not related in any way.
No relation.
Words from English (and other languages) are frequently taken for technical use in mathematics, and sometimes (as here) in two (or more) unrelated ways.