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.