Natural relativized worlds
It seems that there is a confusion in the question between oracles resulting in equality vs. oracles resulting in separation. It is easy to find natural oracles resulting in equality. Oracles for separation is a different matter and your question does not contain an example for them. 

We don't have that many techniques for separating complexity classes and relativized separations are no different. The examples you get are expected to result from those techniques so it is unlikely that such oracles are known.

As far as I know, diagonalization is the only method we know of that can separate non-small complexity classes, so no surprise the oracles separating $\mathsf{NP}$ from $\mathsf{P}$ are such sets.