Chess coaching and coaches training different players What does coaching actually mean in chess?

In relation to that, how is it possible for a chess player to coach two different players? Do the trainees share their work, or does the coach work on different areas of the game with each player, and in that way avoid any "moral conflicts"? 

Two of the most notable examples would be:

-Peter Heine Nielsen, who was coaching Anand for ten years (Wiki), before switching to his main WCC rival Carlsen in 2013 (although he was also helping Carlsen before 2013);

-Vladimir Chuchelov, who coached both Fabiano Caruana and Anish Giri (I am not sure whether he even coached them simultaneously at some point).

On the other hand, Gelfand stated in 2012 that he refused an offer from Kasparov to help him in his match against Anand, since Kasparov was helping Anand in his WC match against Topalov in 2010.

I cannot imagine coaches deliberately trying to keep secret some of the work they have previously done (it seems too artificial). Also, revealing all the secrets of the player they previously worked with, does not seem to me as "morally sound". So, what is the deal then?
Well first lets start with the most simple answer... Chess is not such a simple sport that if taught to multiple people will result in the same thing. Even if what you learned was exactly the same.

If you are interested in getting into more detail then please read on :)

Chess coaching in a sense is just you sharing what you know... That's all there is to it, you tell your students how to think as chess players but that does not necessarily mean they WILL think like that. 

Chess coaches simply hone, and help their students master their skills, they give them the knowledge needed (and of course more) but that doesn't mean it will always result in the same thing.

If I were to give a real life example, this would be the most simple and true to chess.

You can have a machine that takes metal as a material and it constructs a bike, however if you go to another machine and give it the exact same material that doesn't mean it will also produce a bike, it might give you a car! Who knows...

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^ That was the answer, here is some extra stuff if you are interested in more.

Over time in the history of chess, many players like Mikhail Tal, Boby Fischer, Alekhine and so on... All of them had their own unique play style that is remembered until today. However no one can deny the fact that ALL of them had a lot of shared knowledge, they knew the same things yet even so, they played differently. 

Chess is simply a beautiful war on a small 64 square battlefield... Even if you both know the same things and same tactics, you still have your own unique plans.

And now I am done :P Sorry for the long post, but hopefully I helped.