As Péter indicates, this is open. It is related to a question that is a favorite of many. Shelah has conjectured that if $M\subseteq N$ are (proper class transitive) models of $\mathsf{ZFC}$ and $\kappa$ is a cardinal in $M$ with succesor $\lambda$, and $\lambda$ is a cardinal in $N$, then in $N$ we have that $\mathrm{cf}(|\kappa|)=\mathrm{cf}(\kappa)$. Note that if there are two models as you inquire, then $\kappa=\aleph_\omega^M$ contradicts Shelah's conjecture.
Shelah himself proved his conjecture in some cases, namely when, in $M$, $\kappa$ is either regular, or a singular such that $\square_\kappa$ holds. In fact, his proof naturally splits into two parts:
- Under either assumption, in $M$, the combinatorial principle that we now call $ADS_\kappa$ holds.
- If $M\subseteq N$, $(\kappa^+)^M$ is a cardinal of $N$, and $ADS_\kappa$ holds in $M$, then in $N$, $\mathrm{cf}(|\kappa|)=\mathrm{cf}(\kappa)$.
Here, $ADS_\kappa$ is the statement that there is an Almost Disjoint sequence of Sets, in the following sense: A sequences $(A_\alpha\mid\alpha<\kappa^+)$ such that each $A_\alpha$ is an unbounded subset of $\kappa$, and there are functions $g_\beta:\beta\to\kappa$ for all $\beta<\kappa^+$ such that for each such $\beta$, we have that $(A_\alpha\setminus g_\beta(\alpha)\mid\alpha<\beta)$ is a sequences of pairwise disjoint sets.
This appears in the Cardinal arithmetic book, Lemma 4.9 in Chapter VII (page 304), where Shelah remarks that we can also assume instead of 1. that, in $M$, $\kappa$ is singular and $\mathrm{pp}(\kappa)>\kappa^+$.
James Cummings, and later Cummings-Foreman-Magidor have studied Shelah's conjecture. The key references are
James Cummings. Collapsing successors of singulars, Proc. Amer. Math. Soc., 125 (9), (1997), 2703–2709. MR1416080 (97j:03091),
and
James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales and stationary reflection, J. Math. Log., 1 (1), (2001), 35–98. MR1838355 (2003a:03068).
Cummings-Foreman-Magidor prove that rather than square, assuming the weak square principle $\square^*_\kappa$ or the existence of a very good scale suffice. Cummings shows that there are many difficulties in obtaining models contradicting Shelah's conjecture. He also indicates a specific way that one could violate the conjecture and in fact reach your setting, namely, if one has that $$ (\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\aleph_2,\aleph_1) $$ and there is a Woodin cardinal $\delta$. The specific instance of Chang's conjecture needed here is open. It implies that $\{X\subseteq\aleph_{\omega+1}\mid \mathrm{ot}(X)=\aleph_2\}$ is stationary, so we can force with the full stationary tower (for $\delta$) below this set, and $V,V[G]$ is a model of your situation.
Additional later work by Cummings-Foreman-Magidor (on "canonical structure") indicates further difficulties.