"Nothing exists except atoms and the void", and if the physical universe is ultimately reducible to fundamental particles interacting with one another, then the physical properties of those particles ARE the universe. These properties, the rules by which they interact, seem to be statistical and quantized at small scales, and appear to merge into continuous functions in the aggregate at large scales. They are predictable, but in different ways in the two regimes; one by probability functions, the other by real analysis.

If all electrons are identical, then they can be describes by a finite set of numerical constants and functions, even if some of them are statistical, or "fuzzy", in nature. The same can be said for all the other fundamental particles. In other words, it is in principle possible to exactly reproduce any physical system with another made of different particles, given the constraints of quantum physics (the fuzziness). So if any particle will do, then the particles are not critical, it is that list of mathematical properties that defines the system. The universe is nothing but a complex function in some statistical space.

The wrinkle here is that fuzziness, or the quantum indeterminacy of each fundamental particle. It doesn't make sense to speak of identical particles when not even one can be precisely specified because of the Uncertainty Principle. However each particle can be specified by a probability function, and if they are identical then the particles are defined as identical. To use a very crude and probably misleading analogy, two dice are identical if the statistical probabilities of any one number coming up is the same for both. That still doesn't mean that if you throw them both you'll get the same number on each one. But it will occur a certain percentage of the time in the long run. This indeterminacy is essential, or at least, convenient. Without it the universe would be perfectly predictable, at least in principle. But without some constraints provided by the probability function, the universe would be totally chaotic at al scales.

Because of this inherent randomness, even if we could have two small, but identically aranged collections of fundamental particles, they would soon appear different. They woud evolve differently because they weren't identical to begin with (due to the quantum fuzziness). But that is statistical behavior, and it can be modeled statistically. We don't need real particles to reproduce it. A mathematical model would behave like you would expect the system it represented to behave, so you don't need real dice, the model will tell you how often each number comes up, even if it can't tell you in which order.

It is because we have this statistical representation at the discrete micro level, that we get smooth and continuous functions at the macro. Planets move in their orbits under the guidance of simple laws easily represented by algebra and manipulable by differential equations. We don't need to build a mechanical model of the solar system to navigate our spacecraft, we can calculate it. The equations on paper and the ballistic trajectories and gravitational forces are indistinguishable from one another at that scale. And if there is no way to tell things apart, they are, by definition, identical.