Personally I'd like it has something to do with spiritual proximity to the Daedra/Aedra or another divine being.
Souls are power, much like magicka in a sense, given that a captured soul can be used to power a spell. However, access to magicka doesn't mean that a soul will be black in of itself, as shown by Falmer and Spriggans.
My hypothesis is that the distinction comes from how much power the being was imbued with from the gods or whatever you wish to call them.
In the case of Giants, their gods 'died' and they lost part of themselves, devolving into what you see now. And for the Falmer, that connection was severed by the Dwemmer in some sort of experiment.
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It is just one of those things that the more you look at the more complicated it becomes and it depends on how you see it.
Similarly how do you define some races? eg Falmer, Orcs? In TESII Daggerfall Orcs were considered beasts and you could get soul gems filled with Orc souls. In subsequent games you couldn't get one without a black soul gem. So Arkay's Blessing seems to be a flexible thing like Mannimarco mentions. Similarly with the second part (Arkay's Law), perhaps if you interred anything with the proper rites then Arkay's Law would be firm protection from necromancy anyway. The point I am trying to say is that all we know is what people tell us and that is not necessarily true or it can also hide other meanings/interpretations.
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