In "Compositionality in Montague Grammar" (https://pdfs.semanticscholar.org/0b5d/ab9d1718d6ca0c7211c0d81c9a65e4a03759.pdf), talking about classical higher-order logic, Markus Kracht writes that
Montague assumed that.../John/ no longer denotes the individual John but rather the set of properties true of John. In a standard model (where we allow quantifying over all subsets) there is a biunique correspondence between the individuals of the domain and the set of all subsets of the domain containing that individual (such sets are also called principal ultrafilters) (p.12).
He seems to be talking about a standard model of the simply typed lambda calculus, in which quantifiers can range over entities of any type and function types $\alpha \to \beta$ (for types $\alpha, \beta$) are not restricted to a proper subset of the functions from expressions of type $\alpha$ to expressions of type $\beta$. In this context, where $e, t$ are the type of entities and boolean truth values (respectively), the relevant bijection would be between entities of type $e$ (individuals) and entities of type $(e \to t) \to t$ (sets of sets of individuals).
But the cardinality of a set of sets is surely larger than the cardinality of a single individual (the member of all the sets in the principal ultrafilter). So how can there be such a bijection?
I can see how in there is an injection sending every individual in a model to the set of sets containing that individual. But how can there be an injection from the set of all sets containing an individual to that individual, if individuals and sets of sets differ in cardinality?
There can't, and calling this "biunique" is a bit cluncky in my opinion, but I didn't read the article for further context.
Using your notation, you get an injection from $e$ to $(e \to t) \to t$, and that is all that the author claims. Then there is a "biunique" correspondence between the elements of $e$ and the sets-of-sets that are in the image of this injection. That is, if you have a set-of-sets of which you already know that it is the set of all sets containing some individual $x : e$, then in fact this individual is unique (and it is given by the unique element of the intersection of all the sets in the set). But for a general set-of-sets, no such $x$ will exist; just consider a set containing only two disjoint sets, or a set containing no set at all.