What follows uses finiteness of $n$ but I think it works for infinite $n$ too.

First let us note that for finite $n$ the third axiom follows from the first two (because for finite $n$ the third axiom follows from its particular case when $Y_j=X_j$ for all $j$ except one).

So staroids are subsets $f$ of $(\mathscr PU)^n$ whose characteristic functions $(\mathscr PU)^n\to2$ are $\textit{multi-join-homomorphisms}$.

A multi-join-homomorphism $m:\prod_{i\in n}S_i\to S$, where $S_i$ and $S$ are join-semilattices, is a map which is a join-semilattice homomorphism in each argument separately.

It is well known that multi-join-homomorphisms are (at least for finite $n$) representable by the $\textit{tensor product}$ of semilattices. That is, for any join-semilattices $(S_i)_{i\in n}$ there is a join-semilattice $\bigotimes_{i\in n}S_i$ receiving a multi-join-homomorphism $\otimes:\prod_{i\in n}S_i\to\bigotimes_{i\in n}S_i$ which is universal, i. e. for any other multi-join-homomorphism $m:\prod_{i\in n}S_i\to S$ there is a unique (ordinary) join-homomorphism $m_\otimes:\bigotimes_{i\in n}S_i\to S$ with $m=m_\otimes\circ\otimes$.

It follows that staroids may be identified with join-homomorphisms $(\mathscr PU)^{\otimes n}\to2$, which are in turn identifiable with (complements of) ideals of $(\mathscr PU)^{\otimes n}$.

PS

I believe that one may give a more explicit description of $\bigotimes_{i\in n}\mathscr PU_i$ as follows: there must be an isomorphism $\bigotimes_{i\in n}\mathscr PU_i\cong\mathscr P_{\vee\Pi}(\prod_{i\in n}U_i)$, where $\mathscr P_{\vee\Pi}(\prod_{i\in n}U_i)$ is the sub-join-semilattice of $\mathscr P(\prod_{i\in n}U_i)$ generated by the parallelepipeds $\prod_{i\in n}X_i$. Namely, the multi-join-homomorphism $\Pi:\prod_{i\in n}\mathscr PU_i\to\mathscr P_{\vee\Pi}(\prod_{i\in n}U_i)$ sending $X\in\prod_{i\in n}\mathscr PU_i$ to $\prod_{i\in n}X_i$ seems to be universal: for a multi-join-homomorphism $m:\prod_{i\in n}\mathscr PU_i\to S$ a join-homomorphism $m_\Pi:\mathscr P_{\vee\Pi}(\prod_{i\in n}U_i)\to S$ with $m=m_\Pi\circ\Pi$ is uniquely determined by $m_\Pi(\prod_{i\in n}X_i)=m(X)$. It just remains to find out whether this $m_\Pi$ is correctly defined, i. e. whether there are "extra" relations between joins of parallelepipeds in $\mathscr P(\prod_{i\in n}U_i)$...

PPS

Note also that there is a map from $\mathscr P(\prod_{i\in n}U_i)$ to ideals of $\mathscr P_{\vee\Pi}(\prod_{i\in n}U_i)$ sending a subset of $\prod_{i\in n}U_i$ to the set of all finite joins of parallelepipeds contained in it. Thus ideals of $\mathscr P(U^n)$ correspond to (possibly all) staroids.

Realized later:

In fact coproducts of distributive lattices are given by the tensor product, so under Stone/Priestley duality they correspond to cartesian products of spectral/Priestley spaces. Thus $(\mathscr PU)^{\otimes n}$ is the $n$-th copower of $\mathscr PU$ in the category of distributive lattices, hence its dual is the $n$-th power of the dual of $\mathscr PU$, i. e. of $\beta U$. Thus staroids correspond to open/closed sets of the Stone space $(\beta U)^n$.

Explicitly, to an open set $W$ of $\prod_{i\in n}\beta U_i$ corresponds the staroid
$$
\{\ (X_i)_{i\in n}\ |\ \exists (X_i\in\chi_i\in\beta U_i)_{i\in n}\textrm{ s. t. }(\chi_i)_{i\in n}\notin W\ \}
$$
whereas to a staroid $\Sigma\subseteq\prod_{i\in n}\mathscr PU_i$ corresponds the open set
$$
\{\ (\chi_i)_{i\in n}\in\prod_{i\in n}\beta U_i\ |\ \exists (X_i\in\chi_i)_{i\in n}\textrm{ s. t. }\ (X_i)_{i\in n}\notin\Sigma\ \}.
$$

This now works for infinite $n$ (and also for non-constant families $(U_i)_{i\in n}$).