One natural interpretation of the question does give a measurable cardinal. Namely, suppose that $j:V\to M$ is an elementary embedding for which $M$ is an $\omega$-model. More precisely, we have a membership relation $E$ on $M$ and $j:\langle V,\in\rangle\to\langle M,E\rangle$ is an elementary embedding of these structures. Suppose that $j$ is nontrivial in the sense that it is not an isomorphism of every set with the $E$ predecessors of its image. In this case, I claim that there is a measurable cardinal. Fix any set $A$ for which $j$ is not an isomorphism of the elements of $A$ with the $E$-predecessors of $j(A)$. In this case, there must be an element $a\mathrel{E} j(A)$ which is not in the range of $j$. We may define a measure $\mu$ on $A$ by $X\in\mu\iff a\mathrel{E} j(X)$ for $X\subset A$. This is easily seen to be a nonprincipal ultrafilter on $A$. Furthermore, the fact that $M$ is an $\omega$-model will give us countable completeness of $\mu$, for if $X_n\in\mu$ for each $n$, the fact that $\omega^M$ is actually order type $\omega$ implies that every $E$ element of $\omega^M$ is $j(n)$ for some $n$, and from this it follows that the $E$ members of $j(\cap_n X_n)$ are precisely those in every $j(X_n)$. Thus, $a\in j(\cap_n X_n)$ and so $\cap_n X_n\in \mu$. So we have constructed a countably complete nonprincipal ultrafilter on a set $A$, and this implies the existence of a measurable cardinal. (The degree of completeness of $\mu$ is a measurable cardinal.)

**Update.** But there is another interpretation for which the property is weaker than measurability. Namely, it might happen that $j$ is an isomorphism of every set with the $E$-predecessors of its image, but $j$ is not an isomorphism of $\langle V,\in\rangle$ with $\langle M,E\rangle$. Thus, $j$ maps $V$ isomorphically to an initial segment of $M$, but $M$ has new objects at ranks above the ordinals of $V$. So essentially, $V$ is an elementary initial segment of $M$, with $j$ being essentially the inclusion map. This kind of situation can occur in general, since it is possible that $V_\delta\prec V_\kappa$, for example, when $\kappa$ is inaccessible, there are many such $\delta$. But for your question, we would have the situation where $M$ and $E$ are classes in $V$ that $V$ sees to be ill-founded. This is a more subtle situation, but one can build such an example starting only with a weakly compact cardinal, which is strictly weaker than measurability in consistency strength. Suppose $\kappa$ is weakly compact. Let $M_0$ be a transitive set of size $\kappa$, with $\kappa\in M$ and $V_\kappa\subset M$. Since $\kappa$ is weakly compact, there is an elementary embedding $j_0:M_0\to M_1$ to a transitive set $M_1$ of size $\kappa$ with critical point $\kappa$. Applying weak compactness again, we get a map $j_1:M_1\to M_2$, again with critical point $\kappa$. Iterating this, we build $j_n:M_n\to M_{n+1}$ with critical point $\kappa$ each time. Thus, we get an elementary map $j$ from $M_0$ to the direct limit model, which is a structure $\langle N,E\rangle$ that includes $V_\kappa$ in its well-founded part, but which is ill-founded above $\kappa$ and indeed, it has no $\kappa$-th element (since the critical point was $\kappa$ each time, new elements were inserted below any given element of the thread above $\kappa$). The ill-foundedness of the model exists below $j(\kappa)$. Thus, the structure $\langle V_\kappa,\in\rangle$ has an elementary extension to the $j(\kappa)$-rank initial segment of the direct limit. That is, we have $\langle V_\kappa,\in\rangle\prec\langle M,E\rangle$. The structure $M$ and relation $E$ on $M$ have size $\kappa$ and can therefore be coded using subsets of $\kappa$. But now the key point is that since $\kappa$ is inaccessible, we may equip $\langle V_\kappa,\in\rangle$ with all its subsets and still have a GBC model, indeed, a model of Kelly-Morse set theory. This model can see the elementary extension of $V_\kappa$ to the ill-founded model $M$, which is well-founded up to the height of $V_\kappa$. So this is an example of an elementary embedding of the $V$ of $V_\kappa$ into a class model that is an $\omega$-model, but still ill-founded, and is nontrivial in the sense that it is not an isomorphism, but one cannot extract any measurable cardinal, because it was built merely from a weakly compact cardinal.

(See the edit history for my earlier answer, which included some related information, which seems less relevant to me now.)