You should consider the indistinguishability of quantum particles, and the Gibbs Mixing Paradox (entropy from mixing different gases different to mixing containers of the same gas). The degree to which things are interchangeable, and taking the steps to distinguish them, have important real-world consequences on the behaviour of groups of similar constituents. Atoms are distinguishable, unlike their constituents, but in a finite enumerable number of ways, relating to the microstates of the system - that is, they are inbetween being indiscernable, and being unique special snowflakes.
When we roll a dice or flip a coin, we treat the initial conditions as exactly equal, ignore the actual differences in rolls and flips that could allow us to predict outcomes from those; and ignore some possible outcomes like landing on an edge or corner, the rolled object breaking etc. It is a choice to imagine that each roll or flip is indistinguishable, and while they are nearly so, the real objects only approach our idealisation of indistinguishability. Discussed here What is the philosopher's take on information and thermodynamic entropy?
In human morality, when we make statements like The Golden Rule, 'Do as you would be done by', or say 'All men (sic) are born equal', we are also making a choice. In practice, as Orwell noted, we tend to allow that 'some are more equal than others'. But choosing to allow that all humans are 'equal in dignity' once again as a choice has important consequences. In philosophy this interchangeability of moral subjects is termed 'intersubjectivity', and it is foundational to most of our practical moral reasoning about fairness and justice, as discussed here: Is the Categorical Imperative Simply Bad Math? :) And here: Studies exploring the rationale of gender equality I'd relate it to Haidt's Moral Foundations theory, as one of the necessary evolved impulses, to enable deeper human cooperation, especially through language.
Money is a special case of identifying a unit of value for different people and conditions, by specific rules that they agree to share.
In modelling physics with mathematics, we derive rules like conservation of energy from continuous symmetries under transformation. In practice, the gravitational & other fields are slightly different at every point, and fluctuating. But in the regime of spacetime being treated as having a smooth potential, Special Relativity applies, because we have a simple rule of how to swap one thing with another, where those differences can be ignored. Like the difference between one dice toll or coinflip, until we make the effort to distinguish them with less generalised measurements and predictions. I argue here that this is the core metaphor that mathematics is based on: The Unreasonable Ineffectiveness of Mathematics in most sciences
Treating another person's 'point of view' as equivalent, enables a system of signs: 'if I was you I would-', 'if you were me you would-'. This is part of assembling salience landscapes, and deriving narrative modes for causal descriptions. Discussed here: Is the idea of a causal chain physical (or even scientific)?
Symmetry involves microstates, and transformation operations. Do the dart games have symmetries in their statistical variances? Do the chess moves? Playing in the dark shifts the degree the next action can be influenced by the previous action, another kind of mode of symmetry operation. We might reasonably expect the well-rested person can hold a longer series of expected moves in mind, and make decisions in relation to a bigger decision-tree of consequences. Chess builds the significance of each move until the mid-game, when it starts to reduce again. In darts each throw is largely independent of the previous.
So I'd describe 'natural' kinds as ways of reducing the complexity of descriptions, from everything being a unique object, to it having specific symmetries that relate it to other similar objects which relates to making tractable models, and neglecting rare outcomes and small differences to focus on set numbers of possible outcomes, which allows for far great predictability. The success of mathematics on simple systems with few microstates gives it a mystique of being 'out there' of 'objective', when really it is only very similar in system dynamics to many other systems, and so relatively more intersubjective.
Heuristics, and symmetry. Our tools for reducing the complexity of our mental models, in ways that allow us to draw out patterns in our experiences, that we can use in narratives that allow us to make predictions.
