In Harry Potter and the Prisoner of Azkaban, after Harry explained the boggart would change into a dementor for him, Lupin says, "That suggests that what you fear most of all, is fear itself. This is very wise."
Franklin Delano Roosevelt famously said in his first inaugural address (1933), "The only thing we have to fear is fear itself".
Is there a reason for this similarity? Or is it purely coincidental?
The reason for the similarity may simply be that it is a commonplace thought, neither unique to nor original with Franklin D. Roosevelt. The Yale Book of Quotations gives these examples:
Michel Eyquem de Montaigne (1580):
C'est dequoy j'ay le plus de peur que la peur.
The thing I fear the most is fear.
Francis Bacon (1623):
Nothing is terrible except fear itself.
Arthur Wellesley, Duke of Wellington (1831):
The only thing I am afraid of is fear.
Henry David Thoreau (1851):
Nothing is so much to be feared as fear.
Franklin D. Roosevelt (1933):
Let me assert my firm belief that the only thing we have to fear is fear itself — nameless, unreasoning, unjustified terror which paralyzes needed efforts to convert retreat into advance.
user14111 got it right...). I'm not sure that I've ever heard it put quite the way you have quoted it. These are, however, minor quibbles in the context of the question itself.