Mathematicians commonly refer to two objects by the same name if they are equivalent in the given context. In this context, topology, any space that is homeomorphic to a sphere might be referred to as “a sphere” even if literally speaking it’s not a sphere. For instance a topologist would might point at a cube and call it a sphere. In their domain there is no important difference between them so why not?

Also, n-spheres are commonly just called spheres for brevity. So when I say “the fundamental problem of homotopy theory is to compute the homotopy groups of spheres,” I am referring to all homotopy groups of all (n-)spheres simultaneously.

> I don’t see how any of this is limited to spheres.

In fact you’re right, homotopy theory is not just limited to spheres! However, if we could readily compute the homotopy groups of spheres, then we would be able to compute the homotopy groups of any “reasonable space.” Here I’m referring to CW complexes [1] which are a very broad class of spaces that, up to homotopy equivalence, probably includes any space you care to think of. It is for this reason that the problem of computing the homotopy groups of spheres is so fundamental to homotopy theory more broadly.

[1] https://en.wikipedia.org/wiki/CW_complex