Two hydrogen atoms are completely unique to one another...
That statement is false. Quantum mechanics deals explicitly with "identical particles", which are particles that are literally indistinguishable from each other, but are not the same particle. This is an empirically demonstrable violation of the principle of "identity of indiscernibles", which states that if two things are indiscernible any means whatsoever, even in principle, they are the same thing. Even though we have known this principle to be false for almost a century, philosophers still take it seriously for some reason.
There is a relatively simple proof that atoms of the same kind are indiscernible. The heat capacity of solids is a measure of how much the temperature goes up as you add energy to a block of material. The temperature is just the average energy per vibrational mode of the crystal lattice. The number of vibrational modes is intimately linked to the number of distinguisable particles in the crystal. N distinguishable particles have a different number of modes than N indistinguishable particles, so crystals will have a different heat capacity depending on which situation actually obtains.
This can be seen by considering a pair of distinguishable coiins vs a pair of indistinguishable coins. If we have two coins that are distinguished by the labels A and B, we have four ways of arranging them by which face is showing (H for heads, T for tails): AH/BH, AT/BH, AH/BH, AT/BT. If they are not distinguishable we only have three states: H/H, H/T, T/T because there is no way to distinguish AT/BH from AH/BT when we remove the labels.
So by a simple macroscopic measurements like the heat capacity of crystalline solids we can prove positively and directly by experiment that atoms of the same kind are in fact indistinguishable, and that the principle of the identity of indiscernibles is false. It is not false "for quantum particles" but false, absolutely--it just happens that quantum particles are the only case we know of where different particles are genuinely indiscernible from each other. But there is no limited domain of application to this result, and philosopher's attempts to treat it as somehow restrictive to the quantum domain are simply misguided (it turns out that the identity of indiscernibles being false makes nonsense of a bunch of other things philosophers want badly to believe, not least of which is how utterly useless the human imagination is in deciding what is and is not true of the world.)