According to BGS theorem , there is an oracle $A$ such that $P^A\neq NP^A$.
If the relativization operation $B\mapsto B^A$ was a well-defined function, one would expect that from $B^A\neq C^A$ one would be able to conclude that $B\neq C$, e.g. $P\neq NP$ would follow from BGS. However, $P\neq NP$ is still open.
Does that mean that relativization is not a well-defined function?
If so, do we have any example of two provably different relativizations of the same complexity class?
 T. P. Baker, J. Gill, and R. Solovay, "Relativizations of the P =? NP Question"