Joel's answer is spot on, and makes full use of your very demanding definition of an isomorphism of hypergraphs (there are other notions which require the bijection between the ground sets to respect less), but it might perhaps be nice to know that also with both
- less demands on the isomorphism
- more demands on each hypergraphs
the number of isomorphism classes can still reach
$2^{2^{\aleph_0}}$
in natural situations.
For example, in
Infinite Matroids and Determinacy of Games
a proof is given that there exist
$2^{2^{\aleph_0}}$
pairwise non-isomorphic hypergraphs with the additional nice property that each of them is a tame infinite matroid (in the sense of Bruhn--Diestel--Kriesell--Pendavingh--Wollan) on the ground set $\omega$, and each moreover is free of certain minors.
In short, a matroid on an infinite set $E$ is an abstract simplicial complex on $E$ satisfying two additional properties,
an exchange-axiom (analogous to the classical exchange axiom for finite matroids)
and axiom (IM),
which stipulates the existence of maximal elements in certain infinite subposets of the lattice of subsets $(2^E,\subseteq)$, and which does not have an analogue in the theory of finite matroids. If $E$ is finite, (IM) is always satisfied, which is why this notion of infinite matroids extends the classical definition of matroids. The property of being tame means that there does not exist any circuit intersecting a cocircuit in infinitely many ground-set elements. Such matroids naturally arise in the theory of countable infinite graphs.