You write in a comment that metric spaces "don't come up very much in practice".
This is not true (although may reflect the mathematics you have seen so far).
Metric spaces (and, more generally, topological spaces) occur all over the place. I am a working number theorist, and I use the concepts of topology (in all kinds of contexts, sometimes in the context of vector spaces or rings or groups, sometimes in very non-linear contexts) all the time. Geometers and topologists use them still more frequently (perhaps unsurprisingly).
The language and basic results of topology (open and closed sets, continuity, connectedness, compactness) are some of the most flexible and useful concepts in
Added: In my answer I've tended to conflate metric spaces and general topological spaces, but perhaps in your question you want to distinguish them.
(Maybe you are wondering where particular metrics arise that are not induced by
To the extent that geometry is about studying lengths, angles, and related concepts such as curvature, it is very much a subject that revolves around metric spaces, and in modern geometry, geometric topology, geometric group theory, and related topics, many techniques use metrics as the basic strucure.
E.g. Gromov--Hausdorff limits.
E.g. Metric space approach to concepts such as curvature, leading e.g. to
E.g. You might be tempted to think as Riemannian geometry as being a more
analytic than combinatorial/metric-space based subject, because of the role
of differential topology in the foundations. But metric space notions
(such as the two previous examples) are fundamental in modern aspects
of the theory, such as rigidity.