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 WellingtonArthur Wellesley, Duke of Wellington (1831):
The only thing I am afraid of is fear.
Henry David ThoreauHenry 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.