I think the interest is historical. Because mathematics and physics grew up together, physical continuity† could easily be argued to be of mathematical interest & relevance.
† Remember how surprised physicists were when discontinuous "jumps" were found! (The quantum revolution.) iirc this was first proved with black-body radiation experiments.
I can't find the book I'm trying to quote from right now (and will try to edit this later if I remember), but chapter 1 or 0:
Manifolds historically were always understood to be some subset of ℝⁿ. It was only in the early 20th century [let's say due to Noether or Klein] that people decided we could do away coördinate charts altogether, and the notion of an abstract manifold (now the familiar one) was introduced.
You can find this also in Spivak DG1, where he refers to eg pushing forward a tangent bundle as the "modern, clean" way of doing things --- which is less intuitive, but also in a sense preferable, to the older "subset of ℝⁿ" way of seeing things.
You asked if people do study non-smooth spaces. I am not expert enough in any of these to really give an answer, but try googling on:
- pro objects
- p-adics
- paracompactness
- non-Hausdorff
Usually non-Hausdorff would be considered pathological – so I think it would take a beautiful result or another kind of compelling vision to get people interested specifically in some non-Hausdorff object. But one man’s pathology is another’s "very interesting": Dedekind, Cantor, Sierpiński, and Gödel are just a few who had a great time and earned plaudits studying spaces that I consider pathological. AMS obits can be a decent source of information about how mathematicians come to find various spaces / objects interesting. (I have one in mind specifically about Polish spaces, but whose obit it was slips my mind at the moment.)
By "pathological" I mean that I think a lot of mathematicians would be thinking either "Why? That never happens" or "Does anything interesting happen if you start there?"
Finally, so-called "arithmetic manifolds" may be an example in the intersection of mathematics that is taken seriously
∩ not locally Euclidean
. The Weil conjectures; some ideas of Peter Sarnak; Tao-Green theory (arithmetic progressions) ‒ require alternate cohomology theories like étale or crystalline – and within the construction of those there might be an assumption of non-smoothness (eg, the primes are not smoothly distributed). Using less technology, you might be able to google on L-functions (lmfdb.org), weights, or elliptic curves (I believe Silverman has an NSA-crypto style introduction assuming less vocabulary). If you search up "Verdier duality" you will see again people trying to get something like Poincaré duality, but for non-manifolds.
Hope that answers your question! (And assuming others who share it will find this page, in case you aren’t checking the site anymore.)