Has this ever happened to you?

But how can we decide whether this {species/moral theory/consensus node/quantum world} is actually 2 different things, or just 1 thing? Or does it even count at all?

The underlying cause of such dilemmas is that somebody, somewhere in the setup of your ontology and the framing of your problem statement, explicitly or implicitly, made the following move:

For simplicity of analysis, we assume that $X$ is a finite set.

The Promise

This is an incredibly popular simplifying assumption, easily competitive with "we assume this distribution is Gaussian" and "we neglect higher-order terms". It's plausibly the single most popular mathematical modeling assumption in the world, because one needs so little background knowledge to make use of it.

It's also extremely powerful. Here's just some of the mathematical structure you get for free if you accept the assumption:

Computable representation. Saying that $X$ is a finite set can be formalized as $\exists n : N . X ≅ {i : N | i < n}$ , which means that there's some number $k = ⌈ {log}_{2} n ⌉$ such that any element $x : X$ can be losslessly and efficiently represented with $k$ bits.
Metric. Given two elements $a, b : X$ , we say that $d (a, b) = 0$ if $a = b$ , and $d (a, b) = 1$ otherwise. This is well-defined and makes $X$ into a well-behaved metric space.
Compact Hausdorff topology. That metric induces a topology, which is also very well-behaved (in particular, it's compact and Hausdorff, which together make it a compactum).
Symmetry group. The automorphisms of ${i : N | i < n}$ can be transported along the isomorphism to $X$ , so now we have a permutation group, which is useful for formalizing things along the lines of "all elements of $X$ will be treated equally".
Canonical probability measure. By the principle of transformation groups (which I discussed in the middle of this Shortform post about maximum entropy), this permutation group justifies normalized counting measure as an uninformative prior for an uncertain element of $X$ (assuming all elements really should be considered interchangeable).
Ordering. If we have choice as a background assumption, a well-ordering isn't anything special, but from any specific witness of $X$ 's finiteness we get an eminently realizable total order, which is not merely a well-ordering but a completely distrib

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