It is a nice theorem of Zermelo that if we replace the Replacement schema with its second-order counterpart, "The image of a set under any function is again a set", then we necessarily get a model isomorphic to some $V_\kappa$ for an inaccessible $\kappa$.
What happens when we take $\sf Z$, i.e. $\sf ZFC$ without Replacement (but with Separation), and replace the Separation schema with its second-order counterpart: "If $A$ is any collection of sets, then for every $x$, $\{u\in x\mid u\in A\}$ is a set"?
If we add first-order Replacement back, one can see immediately that any model has to be well-founded. Once we have a rank function, an ill-founded model implies there is a set of ordinals of the model without a minimum. Since this set is internally bounded by some ordinal, the second-order axiom implies that this witness is in fact internal to the model.
So in particular, every model of this $\sf ZFC_{1.5}$-hybrid, is isomorphic to a unique transitive model. But this is not all. If $x$ is any set in such transitive model, and $y\subseteq x$, then applying second-order Separation we get that $y$ is also a member of the model. So power sets are computed correctly. Therefore it is necessarily the case that the model is some $V_\alpha$, and since it satisfies $\sf ZFC$ this is necessarily a worldly $\alpha$.
(And easily it can be seen that if $\alpha$ is a worldly cardinal, then $V_\alpha$ satisfies this $\sf ZFC_{1.5}$.)
What happened with the second-order Zermelo (which I will avoid denoting by $\sf Z_2$ for obvious reasons)?
Specifically, since there is no rank function what can we say about this theory?