Which large cardinals have a Matryoshka characterization?

It seems to me that the Berkeley cardinals can be thought of in line with your Matryoshka analogy. Namely, a cardinal $\kappa$ is Berkeley, if for every transitive set containing $\kappa$ there is an elementary embedding $j:M\to M$ with critical point below $\kappa$. Thus, $M$ contains an isomorphic copy of itself, namely the image $j"M$, as a proper subalgebra, which is furthermore an elementary substructure. So this large cardinal notion combines some of the features you mention.

These cardinals are stronger than Reinhardt cardinals in consistency strength, and they are inconsistent with the axiom of choice.

Adding to Joel's answer, there is a weak form of Berkeley cardinals having almost the property you want and whose existence is consistent with the axiom of choice (in fact with ZFC + $V = L$.)

Note: the cardinals defined in Joel's answer are sometimes instead called proto-Berkeley cardinals, but I'll call them Berkeley cardinals here for the sake of consistency. (The difference is whether or not the critical points of such $j$ are unbounded in $\kappa$.)

We may call a cardinal $\kappa$ a virtual Berkeley cardinal if for every transitive set $M$ containing $\kappa$ there is a generic elementary embedding $j : M \to M$ with critical point below $\kappa$, where a generic elementary embedding is defined as an elementary embedding that exists in some generic extension of $V$. (This is a slight abuse of terminology: what is really being defined is the phrase "there is a generic elementary embedding.") This is a type of virtual large cardinal property, other examples being remarkability, virtual extendibility, and the generic Vopěnka principle.

Note that the range of such a generic elementary embedding is a proper elementary substructure of $(M;\in)$ that is isomorphic to $(M;\in)$. However, the embedding and its associated substructure typically exist only in a generic extension in which $M$ (and therefore $\kappa$) is collapsed to be countable.

If $0^\sharp$ exists, then one can show that every Silver indiscernible is a virtual Berkeley cardinal in $L$. This is fairly easy, using the fact that the existence of an elementary embedding $j : M \to M$ with critical point below $\kappa$ is absolute from $V$ to any generic extension of $L$ in which $M$ is countable.

It turns out that a cardinal $\kappa$ is a virtual Berkeley cardinal (as defined above, which should perhaps be called a virtual proto-Berkeley cardinal) if and only if $\kappa \to (\omega)^{<\omega}_2$. Therefore there is a virtual Berkeley cardinal if and only if there is an $\omega$-Erdős cardinal. The forward direction is proved using methods of Silver [1] and the reverse direction is proved similarly to Gitman and Schindler [2, Theorem 4.17].

[1] Jack Silver. A large cardinal in the constructible universe. Fund. Math. 69:93–100, 1970.

[2] Victoria Gitman and Ralf Schindler. Virtual large cardinals. Preprint available at http://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited4.pdf