When I first saw Lion King, I never recognized it as a Hamlet story. In fact, I've never really bought it as a deliberate Hamlet story - I always thought that the similarities were coincidental at best. Not every "Uncle kills the father, son avenges" story is Hamlet.
Well now, with the new 3D release of the movie, we can confirm the answer (courtesy of The Hamlet Weblog):
When we first pitched the revised outline of the movie to Michael Eisner, Jeffrey Katzenberg, Peter Schneider and Tom Schumacher, someone in the room announced that Hamlet was similar in its themes and relationships. Everyone responded favorably to the idea that we were doing something Shakespearean and so we continued to look for ways to model our film on that all time classic.This may or may not be the answer you were looking for. It was not written to be Hamlet. How many "ways to model" their film they found, we don't know.