What are similarities in character and action between Zazu of The Lion King and Polonius of Hamlet?
A major difference between Zazu and Polonius is that the former is an escort, working for Simba's mother, of Nala and Simba as they go play. Polonius is no such escort, and he doesn't work for the side Hamlet's on. Rather, Polonius is a spy for the King, Hamlet's enemy, even though Polonius doesn't know that Claudius is a murderer.
Polonius is more of a major character in the play, unlike Zazu, a stock comic relief character. Polonius is complex and round, unlike the flat and static Zazu. Polonius is an active agent in the plot; Zazu merely comments on it. Polonius is a foil for Claudius, King Hamlet, and Old Fortinbras; he is a father whose death helps his children realize themselves.
Polonius' death is a turning point in Hamlet. It is the first time Hamlet has killed, and Polonius' death will give rise to Laertes' revenge, setting up the fated duel at the end. Zazu is not at the center of the action in The Lion King.
Both of these characters are advisors to the king of the land, and give him support, run errands for him, and are trusted with information. Both Zazu and Polonius are tasked by their kings to look after the son figures, Simba and Hamlet, to make sure that they don't get into any mischief. In short, they are enlisted to spy against the sons, and report any behavior back to the king. Both characters are also rather garrulous, meaning talkative, and a bit silly and laughable at times. It is hard to take them seriously because they take themselves so seriously. One important difference between these two characters is their end--Zazu lives while Polonius does not. Polonius is cruelly murdered by Hamlet in one of the only rash and hasty acts Hamlet performs during the entire play.
It's an interesting question, one that will lend itself to a very interesting paper. Good luck!