Does the Hindley-Milner type system (i.e. STLC with prenex polymorphism) have a category-theoretic model?
Apart from what's already written in the slides you linked to, let me describe one possible approach.
For studying type inference semantically we need a model in which a term can have many types, or none. This naturally leads to Curry-style typing, i.e., we think of $t : A$ as a relation where both the term $t$ and the type $A$ are meaningful by themselves. (The opposite is Church-style typing where a term is always formed together with its type, and it cannot stand on its own.)
We can proceed as follows:

Give a model of untyped terms, such as a model of the untyped $\lambda$-calculus.
Give a model of types.
Define a &quot;has type&quot; relation between terms and types.

A concrete well-known example of this are PER models (although they model entire System F, not just Hindley-Milner):

Consider any model of the untyped calculus, for example, a reflexive domain $D$.
The types are the partial equivalence relations on $D$ (symmetric transitive relations on $D$).
$t \in D$ has type $T$ when $(t,t) \ni T$.

PERs have a rich structure (they form a CCC), for instance the exponential of $S$ and $T$ is the PER $S \Rightarrow T = \{(f,g) \in D \times D \mid \forall (x,y) \in S . (f x, g y) \in T\}$. This is all well-known.
Polymorphism in this model amounts to taking intersection of partial equivalence relations. For instance, $\lambda x . x$ has type $T \Rightarrow T$ for every PER $T$, and so it also has the type $\bigcap_{X \in \mathsf{PER}(D)} (X \Rightarrow X)$, which is the interpretation of $\forall \alpha . \alpha \to \alpha$.
You'll have to find something more nuanced to capture Hindley-Milner without capturing System F. That's what Kammar and Moss set out to do.