The interpretation of higher-order logic depends of course on the set-theoretic background in which it is computed. And if one is willing to change this set-theoretic background, then one may arrive at an answer to the question.

The situation is that whenever one is considering the ultrapower of a given structure $\cal M$ by an ultrafilter $U$ on $I$, then one might simultaneously consider the ultrapower of all other structures $\cal N$ by $U$, and realize that all such ultrapowers and ultraproducts by $U$ fit together in a coherent way, given by the ultrapower of the entire set-theoretic universe by $U$. In particular, one may take the ultrapower of the higher-order structures built from $\cal M$ by applying the power set operation, as indicated by Gerhard. Indeed, one may apply the power set transfinitely.

Specifically, if you have an ultrafilter $U$ on a set $I$, then one may form the ultrapower of the entire set-theoretic universe $V$ by building the structure $\bar V=V^I/U$, consisting of the appropriate equivalence classes $[f]_U$, where $f:I\to V$ is any function on $I$ to set-theoretic objects $f(i)$. Each point $[f]_U$ amounts to the ultraproduct $(\Pi_i f(i))/U$, and in this way each ultraproduct construction is seen as a special case of the ultrapower of the universe (reversing the usual description of ultrapowers as special cases of ultraproducts).

By Los's theorem, the canonical map $j:V\to M$ mapping each object $x$ to $[x]_U$, which is essentially the ultrapower of $x$ by $U$, is an elementary embedding in the language of set theory. This implies, in particular, that $\bar V$ is a model of the axioms of set theory, and furthermore, that any set-theoretic statement about any structure $\cal M$, including higher-order statements of any higher order, including transfinite order, is preserved from the structure $\cal M$ to the image $j({\cal M})$, as interpreted in the new set-theoretic universe $\bar V$. Thus, higher order statements about $\cal M$ in $V$ are directly preserved to the corresponding higher-order statements about the ultrapower of $\cal M$, but interpreted now in $\bar V$.

The interpretation of higher-order logic depends of course on the set-theoretic background in which it is computed. And if one is willing to change this set-theoretic background, then one may arrive at an answer to the question.

The situation is that whenever one is considering the ultrapower of a given structure $\cal M$ by an ultrafilter $U$ on $I$, then one might simultaneously consider the ultrapower of all other structures $\cal N$ by $U$, and realize that all such ultrapowers and ultraproducts by $U$ fit together in a coherent way, given by the ultrapower of the entire set-theoretic universe by $U$. In particular, one may take the ultrapower of the higher-order structures built from $\cal M$ by applying the power set operation, as indicated by Gerhard, and conclude a higher-order analogue of the fact you mention in the question. Indeed, one may apply the power set transfinitely.

Specifically, if you have an ultrafilter $U$ on a set $I$, then one may form the ultrapower of the entire set-theoretic universe $V$ by building the structure $\bar V=V^I/U$, consisting of the appropriate equivalence classes $[f]_U$, where $f:I\to V$ is any function on $I$ to set-theoretic objects $f(i)$. Each point $[f]_U$ amounts to the ultraproduct $(\Pi_i f(i))/U$, and in this way each ultraproduct construction is seen as a special case of the ultrapower of the universe (reversing the usual description of ultrapowers as special cases of ultraproducts).

By Los's theorem, the canonical map $j:V\to M$ mapping each object $x$ to $[x]_U$, which is essentially the ultrapower of $x$ by $U$, is an elementary embedding in the language of set theory. This implies, in particular, that $\bar V$ is a model of the axioms of set theory, and furthermore, that any set-theoretic statement about any structure $\cal M$, including higher-order statements of any higher order, including transfinite order, is preserved from the structure $\cal M$ to the image $j({\cal M})$, as interpreted in the new set-theoretic universe $\bar V$. Thus, higher order statements about $\cal M$ in $V$ are directly preserved to the corresponding higher-order statements about the ultrapower of $\cal M$, but interpreted now in $\bar V$.