If we have a model $N$ and its elementary extension $M$, $N\prec M$ can we consider not only $N$ as a submodel of $M$ but also vice versa $M$ up to an isomorphic copy $M^*$ a submodel of $N$, $M^*\preceq N$ ; is this observation correct? I have come across this issue in other question.
I think so because of the snippet below at least if $M$ and $N$ are elementary equivalent and then if $N$ has cardinality $\lambda$ and $M$ cardinality $\lambda^+$ then we can view $N^*$ as a submodel of $M$ ? Please see the definition 1.0.25 which should mimic saturation in $\lambda^+$ over $\lambda$.
This is certainly not true in general. For instance, $N$ could have greater cardinality than $M$, so there is not even a subset of $M$ of the same cardinality as $N$, let alone an elementary submodule that is isomorphic to $N$.