What is the term for this set
None of your examples holds up: each has queries that allow multiple answers, and each has queries that allow only singleton answers.

So you are really after a property of queries.
On the most abstract level, given a set $S$ the term you are looking for is function:

$\qquad f : Q \to S \\
\qquad f(q) = s \text{ s.t. } q(s) = 1$

where $Q$ is the set of queries (predicates), i.e. $Q \subseteq S \to \{0,1\}$. Note how $f$ is a (well-defined) function if and only if all $q \in Q$ have a support (set of values on which $q$ is $1$) of size $1$.

If queries end up being true for (i.e. returning) multiple values, we call $f$ a relation. Or, alternatively, a function that maps queries to (possibly empty) sequences of results, i.e.

$\qquad f : Q \to S^*$.