Suppose I have a set $S$ containing $n$ bit strings, where $n$ is on the order of about 10. Consider
$$\mathfrak{S} = \{ R : R \subseteq S, |R| \geq 2 \},$$
the collection of subsets of $S$ with two or more elements.
Does there exist a hashing function which outputs the same hash for any member of $\mathfrak{S}$?
A potential relaxation in which we order the elements of the subsets in $\mathfrak{S}$ would be acceptable as well.
Does there exist a hashing function which outputs the same hash for any member of $\mathfrak{S}$?
One can certainly define such a hash function.
If we assume $\text{SHA}$ is a standard secure hash function, then one simple $H$ that means your requirements is:
$$H(x) = \begin{cases} \text{SHA}(0) & \quad \text{if } x \in \mathfrak{S} \\ \text{SHA}(1 | x) & \quad \text{if } x \not\in \mathfrak{S} \end{cases}$$
$H$ is a secure hash function, except that it hashes two elements of $\mathfrak{S}$ to the same value; that is, if someone finds a collision in $H$ (not involving two elements of $\mathfrak{S}$), one can immediately find a collision in $\text{SHA}$, which we assumed was secure.
