This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse.
Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ of an inaccessible cardinal $\kappa$ the unique forcing notion (up to forcing equivalence) which is $\kappa$-c.c., has a dense set of size $\kappa$ and which collapses all infinite cardinals below $\kappa$ to $\omega$?
An analogous fact is known and often used in the case of $\text{Coll}(\omega,\theta)$, which is the unique forcing notion of size $|\theta|$ necessarily collapsing $\theta$ to $\omega$, and a proof of this can be found for example as (folklore) lemma 18 of my paper Structural connections between a forcing class and its modal logic, and elsewhere.
