A previous question, Principal ultrafilters, asked for clarification of Markus Kracht's assertion that "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 here: https://user.phil-fak.uni-duesseldorf.de/~rumpf/SS2010/ComSem/Kra08.pdf).
The OP wished to know whether the qualification "where we allow quantifying over all subsets" was necessary, and why a standard model was necessary for the generalisation to hold.
Noah Schweber responded that "defining the map "Send x to the set of all sets containing x" requires us to quantify over sets of sets. It's not that this bijection would fail to 'work' in too weak a logic, but that $\textit{it would fail to exist."$\hspace{0.2cm}($my italics})$
Would such a function fail to exist in a second order logic with the henkin semantics simply for the banal reason that there may be subsets that $x$ belongs to but which are not part of the model since Henkin models are restricted to a proper subset of all sets?
Just what was meant by fail to 'exist' in this sense?
A further question: given it is standard for people working in the Montagovian tradition of formal semantics for natural language to treat names of individuals such as $\textit{John}$ as denoting the principal ultrafilters they generate, is this practice at all put into question if we are using a Henkin semantics and not the full semantics (and so can't gather together all subsets of the domain to which $John$ belongs)?
The issue is not really the quantification over sets. The map that is being constructed is between individuals ("first order" or "type 0" objects) and sets of sets of individuals ("third order" or "type 2" objects). So the map itself can't be constructed in second-order logic regardless of the semantics.
The issue is that, in non-full models, there might be two distinct individuals which cannot be distinguished by any set in the model - each set in the model contains both, or neither, of the individuals. Then the map that is being constructed from individuals to sets of sets of individuals will no longer be injective.