A man of many hats

Billy and Willy are alike, identical to their bellybuttons.
But they are not rich. In fact, they’re quite poor.
(And somewhat mad.)
Come Christmas time, each puts his hat in a gift box and sticks it under the tree.
They exchange gifts this way every year at Christmas.
Year after year, using the same box, same bow and ribbon.
Billy and Willy open their gift box together, smile widely, and say “It’s exactly what I need!” in chorus.
They died on the same day, were buried in the same coffin.
With the one hat on William’s head, for all eternity.