If $M$ is a manifold and $x \in M$, then the ring of smooth germs in $x$ is canonical isomorphic to the localization $C^{\infty}(M)_{\mathfrak{p}}$, where $\mathfrak{p} = \{f \in C^{\infty}(M) : f(x) = 0\}$. I believe this was known long before the Zariski topology. And yet you get the same message: Geometric localization is expressed algebraically in introducing inverses for functions for which it makes sense, thus you can call it localization.

I'm also interested in a historical source, but I don't think that the terminology emerged from algebraic geometry. It's at least one instance for motivating this terminology, among othes such as differential geometry and also functional analysis.