= Solution
The edge choices are understood to be independent, and $r$ is a fixed positive integer. We prove the <hypergraph extension property> and then show directly that it determines a unique countable <hypergraph isomorphism> type.
For a finite $S\subseteq\mathbb N$, prescribe a family $\mathcal A\subseteq\binom S{r-1}$. We want a new <vertex> $v$ such that
$$
F\cup\{v\}\text{ is a hyperedge}\quad\Longleftrightarrow\quad F\in\mathcal A,
\qquad F\in\binom S{r-1}.
$$
For any $v\notin S$, the <probability> of precisely this pattern in the <countable random uniform hypergraph> is
$$
q=p^{|\mathcal A|}(1-p)^{\binom{|S|}{r-1}-|\mathcal A|}>0.
$$
Two different candidate vertices use disjoint <hyperedge> decisions: all the other vertices in each tested edge lie in $S$. The successes are therefore <independent random variables> with the same positive <probability> $q$. The <probability> that the first $N$ candidates all fail is $(1-q)^N\to0$. The same argument after discarding any finite initial set of candidates shows that infinitely many witnesses exist <almost surely>.
There are only countably many finite subsets of $\mathbb N$, and finitely many prescriptions for each. A countable union of null events is null. Consequently the random <uniform hypergraph> satisfies every such extension prescription simultaneously <almost surely>.
Now let $H$ and $K$ be two countably infinite $r$-<uniform hypergraphs> with the <hypergraph extension property>. Enumerate their <vertex sets> and construct increasing finite partial <hypergraph isomorphisms> by the <back-and-forth method>. At a forward step, take the least unused vertex $u$ of $H$. Its incident <hyperedges> with each $(r-1)$-subset of the current domain specify a pattern on the matched vertices of $K$. The <hypergraph extension property> supplies an unused vertex $v$ of $K$ with exactly this pattern. Adding $u\mapsto v$ preserves and reflects every <hyperedge> in the enlarged domain. At the next step, take the least unused vertex of $K$ and perform the identical construction with $H$ and $K$ interchanged.
The union of the finite maps is defined on every vertex and has every vertex in its range. It is a <bijection>, and each finite $r$-set appears at some finite stage, so all <hyperedges> and nonedges are preserved. Thus it is a <hypergraph isomorphism>. Existence of at least one structure with the property follows from the probability-one construction already proved; fix one representative and call it $G^{(r)}_{\mathrm{univ}}$. Therefore
$$
\boxed{G^{(r)}_{\mathbb N,p}\cong G^{(r)}_{\mathrm{univ}}\quad\text{almost surely},\qquad0<p<1.}
$$
The <hypergraph isomorphism> type does not depend on $p$. In the case $r=2$ it is the <Rado graph>. For $r=1$, the argument says that the infinitely many selected and unselected singleton vertices determine the same one-uniform structure for every $p\in(0,1)$.
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