It's called the Manhattan Bridge because you can see the fancy arch and hangout in the promenade only on the Manhattan side. When you hit Brooklyn, there's nothing fancy there. All you have is pavement and an underpass. And there's other bridges too: Williamsburg, Washington, Brooklyn, and the Edward Koch.
It's not the only bridge in Manhattan. But I'm not sure really regarding the name. Brooklyn Bridge is not that far from it.
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