The people claiming Gustave assumed Zero's ID and is, in fact, F. MA
...are overlooking an important clue that shows they are wrong.
When Jude Law's character first sees F. Murray Abraham in the lobby and asks the Concierge about him, he gets told that's Mr Moustafa.
At that point, they show a couple of newspaper clips that are 4 years apart - one from 1936 and the next from 1940.
The encounter with the soldiers on the train after the will dispute resolution happens in 1932. So if Zero got shot then, there is no way those newspaper clippings, with clear pics of younger Zero, would be possible over the next 8 years (that show him as the heir to Gustave's fortune and taking over the hotel). Given how well known he was, as per the author's narration, we can assume there were many more newspaper articles over the years about Zero's fortunes, life and what not - something that Gustave would have no way of pulling off.
I think Zero is supposed to be older than the actor playing his younger self appears to be. And in the spa scene when FMA talks to Jude Law, he doesn't look very white - he is quite tan and almost looks Arabic. So F Murry Abraham is indeed Zero, not Gustave.