According to wikipedia,
...the Löwenheim–Skolem Theorem states that for every signature $σ$, every infinite $σ$-structure $M$ and every infinite cardinal number $κ ≥ |σ|$, there is a $σ$-structure $N$ such that $|N| = κ$ and
- if $κ < |M|$ then $N$ is an elementary substructure of $M$;
- if $κ > |M|$ then $N$ is an elementary extension of $M$.
Now let $\kappa$ be inaccessible. Then $V_\kappa$ agree with the ambient universe about $\aleph_1,$ furthermore $V_\kappa$ knows that $\aleph_1$ is uncountable. So by Lowenheim-Skolem, we can find a countable elementary substructure $M$ of $V_\kappa$. Since $V_\kappa$ is well-founded, so too is $M$. Hence, we can collapse $M$ to obtain an (isomorphic) transitive model $T$. But since $T$ is countable, hence $\aleph_1^T$ does not equal $\aleph_1.$ Thus certainly, it is not the case that the inclusion $T \hookrightarrow V_\kappa$ is an elementary embedding. Yet there exists an elementary embedding $T \hookrightarrow V$.
This seems kind weird. Can anyone explain what's going on here?
This is known as the Skolem Paradox (often used with $\Bbb R$ rather than $\aleph_1$, but the principle is the same).
If you're willing to accept an inaccessible cardinal, then you are willing to accept far weirder things than "we can find other embeddings which are not inclusion".
$M$ and $T$ are isomorphic, but they are not the same. Incidentally, if you replace each element $x\in V_\kappa$ by $x\cup\{V_\kappa\}$ then you still have a natural way of identifying the resulting set with $V_\kappa$, so it is a model of $\sf ZFC$, but suddenly... the empty set is not empty.
Why is my example preposterous and the Skolem paradox isn't? Because mine is a hyperbole, of course. But the idea is the same. There is usually more than one way of doing things, especially when things are infinite.
If you want to keep insisting that something here is wrong, then you need to reject first-order logic altogether. But then you run into other problems.