How should I describe the relationships between type expressions?
The set-theoretic intuitions can make sense in the semantics, especially in the context of realizability semantics (where types are interpreted as sets of terms). In this case, the polymorphic type $\forall \alpha. \mathrm{Maybe}(\alpha)$, written Maybe a in Haskell, intuitively corresponds to the intersection over every type $T$ of the
elements of $\mathrm{Maybe}(T)$. Or in other words, $⟦\forall \alpha. \mathrm{Maybe}(\alpha)⟧ = \bigcap_T ⟦\mathrm{Maybe}(T)⟧$, where $T$ ranges over the semantic domain (and elements of the semantics are injected into the syntax).

On the other hand, set-theoretic intuition does not really make sense on the side of the type system, because standard set operations are not allowed. For instance, it is not possible to refer to the intersection of two types in the syntax (at least when you are not working with intersection type systems, and I don't think this is the case in Haskell). However, one operation on types that is primitive in all type systems with polymorphism is instantiation. This corresponds to the elimination rule for the universal quantifier (or polymorphism), and it states that if you have a term $t$ of type $\forall \alpha, \mathrm{Maybe}(\alpha)$ then the term $t$ can be seen as an element of type $\mathrm{Maybe}(T)$ for any type $T$, for example $\mathrm{Integer}$.