Yes, the mapping preserves area of any shape. You can convince yourself of this by taking by small patches on the sphere, between two constant latitude lines and two longitude lines, which I believe is what they did with the state of Colorado and the sate of Wyoming. Anyway, any nice enough shape is made up, to sufficient accuracy, by a large number of these curved rectangles, and these quite definitely are mapped in an area-preserving manner. This is the oldest example of a "symplectic" map.
I do not know that much about the history of this exact example, but I do know that a book of Archimedes called The Method was thought to be lost until about 1900, and translations are available. See EUDOXUS and METHOD and SPHERE_AND_CYLINDER finally MOOSE_AND_SQUIRREL
Alright, somebody at Wikipedia is not paying attention. The equal area MAP projection is due to Archimedes.