As an example of (B), I'd mention Connes's classification of injective (type $II_{1}$) factors from the theory of von Neumann algebras. In this case, many apparently disparate constructions turn out to give a single object.

Classification of injective factors. Cases II1,$ $II_\infty,$ $III_\lambda,$ $\lambda \not=1$. Ann. of Math. (2) 104 (1976), no. 1, 73-115.