If we start with a model of $\sf ZFC$, $M$ and $(P,\le)\in M$ is a notion of forcing, $G\subseteq P$ a generic filter, then in $M[G]$ we can define some generic object from $G$. For example, if $P$ is the Levy collapse of $\omega_1$ to $\omega$ then $G$ defines a new function $f\colon\omega\to\omega_1$ which is bijective.
Now suppose that we have a product forcing $P=\prod P_i$ in $M$, then the generic filter $G$ can be projected on every coordinate and $G_i$ (its projection) is a generic filter over $P_i$, which defines some generic object. Then a priori we can think that $G$ defines some generic collection $\{g_i\}$ such that $g_i$ is the generic object defined by $G_i$.
So for example, if we take the product of two Cohen-like forcings, one adding a subset of $\omega$ and the other adding a subset of $\omega_1$ - we can think of the collection as the pair of the new subsets.
In Jech Set Theory, 3rd Millennium edition, in the relevant chapter (Ch. 15) Jech discusses this very shortly, proving some basic theorems about this. However in the exercises there is only one problem related to this issue:
Let $P$ be the notion of forcing (15.1) that adjoins $\kappa$ Cohen reals. Then $P$ is (isomorphic to) the product of $\kappa$ copies of the forcing for adding a single Cohen real (Example 14.2).
This means, that we can think of the product of $\kappa$ Cohen forcings as adding $\{g_i\mid i<\kappa\}$ as a set of $\kappa$ new Cohen reals, just like we would think at first.
However, there is no mention of this being true or false in a general framework. So to my question:
Suppose $P=\prod P_i$ is the product of $\kappa$ copies of some $P'$ a fixed notion of forcing, can we automatically assume that $G\subseteq P$, a generic filter, adds a set of $\kappa$ new generic elements, each defined by a generic filter, $G_i$ over $P'$?
If this is true, then we can ask even further:
Suppose $P=\prod P_i$ is a product of $\kappa$ notions of forcings, can we say that $G\subseteq P$, a generic filter, adds a set of generic objects each defined solely by $G_i$?