"Classification" might be a very strong word for this example, but I think real quadratic polynomials underwent something like the development you describe. They are classified by their discriminant: positive for two distinct roots, zero for a double root and negative for no roots ("three families"). The generalization of complex polynomials and roots simplifies this: discriminant zero for a double root and non-zero for two distinct roots ("two families"). When you specialize this to real polynomials you end up with what you started with, with the added bonus of two complex conjugate roots instead of none in the case of a negative discriminant.

Surely this is only a toy example, but it does show that broadening the viewpoint can reduce the number of cases to consider.