What is a conventional name for a set of values having no properties except that values are distinct? I know essentially nothing about math but I'm interested in very low-level concepts.
I'm thinking of something like a finite or infinite set (although I'm not married to consider sets per se, maybe some relevant stuff is from category theory or type theory or whatever) such that the only thing one could do with two variables having the values from the set is compare whether they have identical values or not. Nothing else about these values is defined: no ordering, no difference, no ratio, no addition/multiplication, no builtin relationship to the naturals or whatever, etc.
What are some names for this kind of thing that I can use to find more reading about it and how it relates to the rest of math or logic?
 A: They're called abstract sets.
Personally, I would simply call this a "set," or perhaps a "mere set" or "unstructured set" to emphasize that there's no further structure around.
A: A set is a collection of distinct elements in which order doesn't matter.  A multi-set is a collection of elements (with possible repitition) in which order doesn't matter.
Neither of these (or the next terms below) require or need any operations to be defined for them.  For example, $\{ziggyswooglehorn, 5, \color{#C00}{\rm Red}\}$ is an example of a set with the elements $ziggyswooglehorn$, $5$, and $\color{#C00}{\rm Red}$ contained in it.  There is no mention however of what operations apply to or information you can extract about the elements in question except for their equality or inclusion in the set.  That is not to say that you cannot define such operations, it is just that they are not yet defined.
Similar concepts include posets (a set of distinct elements with partial order), ordered sets (a set of distinct elements with order), and sequences (an ordered multiset).
We could also consider the set $\mathbb{Z}=\{\dots,-2,-1,0,1,2,3,\dots\}$.  Without more information, we cannot say if there is in fact any operations that we can use with it.  If we wish to consider a set with operations and orders if the operations and/or order is not implicitly understood, we would write $(\mathbb{Z},+,\times,\leq)$, or something like $(A,\spadesuit,\heartsuit,\smile)$, where we can define also what it means to do $x\spadesuit y$.
A: The technical term for a set of values having no properties except that values are distinct is "set".
