  We prove that the multi-Rees algebra $\mathcal{R}(I_1 \oplus \cdots \oplus
I_r)$ of a collection of strongly stable ideals $I_1, \ldots, I_r$ is of fiber
type. In particular, we provide a Gr\"obner basis for its defining ideal as a
union of a Gr\"obner basis for its special fiber and binomial syzygies. We also
study the Koszulness of $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ based on
parameters associated to the collection. Furthermore, we establish a quadratic
Gr\"obner basis of the defining ideal of $\mathcal{R}(I_1 \oplus I_2)$ where
each of the strongly stable ideals has two quadric Borel generators. As a
consequence, we conclude that this multi-Rees algebra is Koszul.
