I'm trying to understand types better in model theory and, in particular, how the set of all types interacts with various ways of building new models out of old ones (e.g. the Cartesian product $N \times M$).
What do we know about the set of all types and the set of all realizable types for a binary Cartesian product of models?
Suppose $M$ is a structure. Let $\mathrm{TY}(M)$ denote the set of all $n$-types of $M$ over any subset of $M$ and for any value of $n$.
Let $\mathrm{RT}(M)$ denote the subset of $\mathrm{TY}(M)$ consisting of exactly the realized types. For all types $p(x)$, $p(x)$ is in $\mathrm{RT}(M)$ if and only if $p(x)$ is in $\mathrm{TY}(M)$ and there exists some $b \in M^n$ such that $M \models p(b)$.
Fix a signature $\sigma$. Let $M$ and $N$ be structures over $\sigma$.
Let $M \times N$ be the Cartesian product of $M$ and $N$, with the interpretations of functions and relations defined componentwise as usual.
I'm interested in whether the following are true. Assuming they're false, I'm interested in what we do know about the set of all types and the set of realizable types in this setting.
$$ \mathrm{TY}(M \times N) = \mathrm{TY}(M) \cap \mathrm{TY}(N) \tag{1} $$
$$ \mathrm{RT}(M \times N) = \mathrm{RT}(M) \cap \mathrm{RT}(N) \tag{2} $$
I can prove the first claim.
For an ordinary parameter-free sentence $\varphi$, $M \times N \models \varphi$ holds if and only if $M \models \varphi$ holds and $N \models \varphi$ holds. This falls out of the way that the interpretation of function symbols and the interpretation of relation symbols are defined in $M \times N$.
For a sentence with paramters $\varphi(A)$, the same argument holds. If $\varphi(A)$ holds in $M \times N$, then it holds in $M$ and $N$ individually. Also, every possible pair $(m, n)$ is in $M \times N$, so if $\varphi(A)$ holds at some $m$ in $M$ and at some $n$ in $N$ for each parameter, then we can build a the values of the parameter $(m_1, n_1), (m_2, n_2) \cdots$ and each $(m_i, n_i)$ will be in the domain of $M \times N$.
If this holds for every sentence individually, then $p(x)$ is finitely satisfiable in $M \times N$ if and only if it is finitely satisfiable in $M$ and finitely satisfiable in $N$ because finite satisfiability can be assessed by only ever examining one sentence at a time.
For realizable types, I'm a lot less certain. Is there a way to build the set of realizable types over the Cartesian product of $M$ and $N$ out of the set of realizable types for each structure individually?
Contra your claim, (1) isn't true at all. Already at the parameter-free level we have a problem: Cartesian products do not preserve arbitrary sentences. For a trivial example, take two finite structures each of size $>1$. For a nontrivial example, consider the standard exercise that $\mathbb{Z}\not\equiv\mathbb{Z}\times\mathbb{Z}$ as groups. And when we bring parameters into the picture, the statement doesn't even parse since $M\times N\not\subseteq M\cap N$ in general.
Unfortunately I don't see offhand any good sense in which the set of types of $M\times N$ can be built from the sets of types of $M$ and of $N$ separately, let alone in which realization plays well with this.