Is there a Turing Machine that can distinguish the Halting problem among others? Can there be a Turing machine, that given two oracles, if one of them is the Halting problem, then this machine can output the Halting problem itself? 
Clearly, if the first oracle is always the Halting problem, then such machine exists, just copy the first oracle. But if the Halting problem can either be the first or the second oracle, can a Turing machine distinguish which one is the Halting problem (for all such pairs of oracles)?

No. Using programs where both oracles give the same output, you obviously cannot distinguish them. Therefore the only way to distinguish the oracles is to analyse programs on which both oracles give different results.
However, if the two oracles give different results, the only way to decide which oracle is the one answering the halting problem is to determine whether that algorithm halts.
Since your Turing machine is intended to answer the question for any pair of oracles, it has to be able to answer that question for any arbitrary algorithm, in order to distinguish the oracle that gives a different answer at only that one algorithm from the halting oracle.
But that means, quite literally, that the Turing machine must be able to solve the halting problem, which we know is impossible.
