You've got it backwards: in this case, if a piece can rotate, all of its possible positions count as the same configuration. In other words, the problem is posed in such a way that we can ignore the fact that pieces can (sometimes) have the freedom to rotate.
If we only allow the pieces to mate in the official manner, at right angles, then it’s still a combinatorics problem with an finite integer answer. There’s an extensive tradition of picking out discrete subfamilies out of continuous things and studying them with discrete tools—symmetries of polygons, tilings, regular polyhedra, crystalline lattices, etc., are all in that group (no pun intended).
On the other hand, just because your problem sounds discrete doesn’t mean that the continuous toolkit isn’t going to be useful for it, as the inordinate utility of generating functions[1] (closely related to Fourier transforms) shows. The other way around also works, with the theory of smooth symmetries (Lie groups) making good use of the discrete things I mentioned above.
It’s all a single field, as Bourbaki wanted to point out by ungrammatically naming their course Éléments de mathémathique (not -es). Even if they omitted some significant parts of that fields that they didn’t know properly or weren’t well-developed yet (e.g. logic counts as some of both).
