What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.)

If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, the existence of an $\omega$-model has consistency strength strictly weaker than the existence of a transitive one. (This follows, for instance, from absoluteness considerations: Any transitive model of $\mathsf{ZF}$ has as elements some $\omega$-models.)

It is true that there are $\beta$-models whose ordinals do not form an initial segment of the true ordinals, even if the membership relation of the model is true membership. For instance, if $V_\alpha$ is a model of set theory, consider any of its countable elementary substructures.

Note that models whose membership relation is true membership are $\beta$-models, so they are isomorphic, via the collapse, to transitive models. The point of the Mostowski collapse is that there is a natural rank that we can assign to the elements of a standard model $M$, so that if $M\models x\in y$ then the rank of $x$ is strictly smaller than the rank of $y$. (If the membership relation of $M$ is true membership, we can use as rank the true set-theoretic rank of the sets in question.) A straightforward transfinite recursion then allows us to replace the elements of $M$ by copies that form a transitive model, simply by defining $\pi:M\to V$ by $\pi(y)=\{\pi(x)∣M\models x\in y\}$. Choice plays no role here.

What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.)

If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, the existence of an $\omega$-model has consistency strength strictly weaker than the existence of a transitive one. (This follows, for instance, from absoluteness considerations: Any transitive model of $\mathsf{ZF}$ has as elements some $\omega$-models.)

It is true that there are $\beta$-models whose ordinals do not form an initial segment of the true ordinals, even if the membership relation of the model is true membership. For instance, if $V_\alpha$ is a model of set theory, consider any of its countable elementary substructures.

Note that models whose membership relation is true membership are $\beta$-models, so they are isomorphic, via the collapse, to transitive models. The point of the Mostowski collapse is that there is a natural rank that we can assign to the elements of a standard model $M$, so that if $M\models x\in y$ then the rank of $x$ is strictly smaller than the rank of $y$. (If the membership relation of $M$ is true membership, we can use as rank the true set-theoretic rank of the sets in question.) A straightforward transfinite recursion then allows us to replace the elements of $M$ by copies that form a transitive model, simply by defining $\pi:M\to V$ by $\pi(y)=\{\pi(x)∣M\models x\in y\}$. Choice plays no role here.

What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.)

If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, the existence of an $\omega$-model has consistency strength strictly weaker than the existence of a transitive one. (This follows, for instance, from absoluteness considerations: Any transitive model of $\mathsf{ZF}$ has as elements some $\omega$-models.)

It is true that there are $\beta$-models whose ordinals do not form an initial segment of the true ordinals, even if the membership relation of the model is true membership. For instance, if $V_\alpha$ is a model of set theory, consider any of its countable elementary substructures.

What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.)

If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, the existence of an $\omega$-model has consistency strength strictly weaker than the existence of a transitive one. (This follows, for instance, from absoluteness considerations: Any transitive model of $\mathsf{ZF}$ has as elements some $\omega$-models.)

It is true that there are $\beta$-models whose ordinals do not form an initial segment of the true ordinals, even if the membership relation of the model is true membership. For instance, if $V_\alpha$ is a model of set theory, consider any of its countable elementary substructures.

Note that models whose membership relation is true membership are $\beta$-models, so they are isomorphic, via the collapse, to transitive models. The point of the Mostowski collapse is that there is a natural rank that we can assign to the elements of a standard model $M$, so that if $M\models x\in y$ then the rank of $x$ is strictly smaller than the rank of $y$. (If the membership relation of $M$ is true membership, we can use as rank the true set-theoretic rank of the sets in question.) A straightforward transfinite recursion then allows us to replace the elements of $M$ by copies that form a transitive model, simply by defining $\pi:M\to V$ by $\pi(y)=\{\pi(x)∣M\models x\in y\}$. Choice plays no role here.