We demonstrate that the direct sum of ideals satisfying the strong $\ell$-exchange property is of fiber type. Furthermore, we provide Gröbner bases of the presentation ideals of multi-Rees algebras and the corresponding special fibers, when they are associated with an $n$-dimensional Ferrers diagram that is standardizable. In particular, we show that these blowup algebras are Koszul Cohen--Macaulay normal domains and classify their singularities.