I think MLK had in mind the JudeoChristian culture. And back then, culturally, blacks and whites did act a lot more like one another than now. At that point, blacks in this country had been living in Western civilization for 350 years. All people dressed more or less the same way, according to what they could afford. People who were small-time farmers and mom'n'pop businesses could live roughly similar lives, if separate by territory. People who attended religious services gave a more or less similar appearance -- dresses and suits, hats and gloves. Racism and socioeconomic restrictions revolved around territorial, microcultural and reproductive instincts below the surface. MLK wanted Christian men and women to be treated with respect and equal opportunity economically. To my knowledge, he did not get into the topic of cross-reproduction openly; however, this was an expressed fear by white separationists, who then as now, were horrified by the behavior of the transgressive behaviors of those most different from themselves, while tolerating it from those most like themselves.
Black Jews in New York City, early 1900s
Baptist church dinner, Annapolis MD 1943
Asbury Methodist Church, Natchitoches, Louisiana, 1950s