Yes.

I'll give you the two ends of the spectrum. Nazi Germany itself gleefully pointed out that hypocrisy in its propaganda directed at American soldiers, and the National Association for the Advancement of Colored People pointed out the hypocrisy while trying to advance the status of black Americans.

If you read the memoirs of American soldiers or scholarly accounts of German propaganda, you'll find that the idea of "race mixing" pops up again and again. In propaganda radio broadcasts and loudspeaker broadcasts, the German propaganda arm loved the refrain that "while you are fighting and dying your girlfriend/wife is having relations with a black/jew/4F man back in the United States."

The Calvin College German propaganda page is a useful tool in looking at examples of WWII and East German propaganda, and I recall several issues of the magazine Signal that deal with the issue of race-mixing. There's a fascinating paper dated from 1942 and intended for the German propaganda-makers themselves that applies German racial theories to the United States. If you get a chance, read it.

Back on the home front, the NAACP was lobbying President Franklin Roosevelt hard for measures to ensure equality. Roosevelt was hampered from pursuing equality by the need to preserve his political coalition, which included segregationist Southern Democrats. In order to ensure political tranquility, he felt he could not desegregate the armed forces. What he could do was issue Executive Order 8802, which prohibited discrimination in war production. Even this measure, however, was contested by Southern Democrats, and there were exceptions to the order.

The order had the odd effect that civilian contractors frequently received better treatment than black soldiers did. Studs Terkel's The Good War has some excellent passages dealing with the racial issues that black soldiers faced. In interviews, Terkel collected the memories of men who saw German POWs receive better rations and care than black American soldiers in Europe. Black soldiers received worse rest and recreation facilities, worse training facilities, and on occasion, even worse medical care.