"Stray", Benjamin Rosenbaum and David Ackert

SS F&SF (Fantasy & Science Fiction) December 2007 - fantasy
In the era of FDR, Ivan, a down and out immortal, is found by a Muriel, a colored doctor's daughter. For her, Ivan becomes a light-skinned black man. They marry and Ivan works to fit into Muriel's world. Then, in a cemetery, Ivan meets crying white girl.

This story seems profound, but I'm a bit fuzzy about the point it's trying to make. Sure, there's the race issue. My guess is that this story is about making people like you.