I think this a difficult question to answer mainly because each individual soldier would handle their own experience. Institutionally? Not really. Well, at least not anytime quickly after the war. Truman did desegregated the armed forces a full three years after the war. So better late than never.
During the war, the military segregated the units, with African-American units kept separate while largely relegated to supply and logistics jobs (which would cause issues while in France-so it wasn't as if Europe was some utopia for Black soldiers). White officers-mostly Southerns because the military thought that Southern white men would be the best people to "handle" Blacks, led the African-American.
One example of the discrimination that Black soldiers faced, despite officers and enlisted serving in combat together, is the fact not a single Black soldier was awarded the Medal of Honor. In the 90s the government commissioned a report, the Shaw Report, to find out why:
The Army entered World War II with racial assumptions about the inferiority of black soldiers as combat troops ... ; these assumptions dominated Army thinking, supported the policy of segregating African Americans in separate units and underlay the policy which relegated blacks overwhelmingly to service in noncombatant units.
Regarding racial relations:
It must be understood that a pattern of racial prejudice on the part of the white leadership of the 92nd Division pervaded the experience, contributed to the very low morale of that unit and poisoned the relationships between the senior officer leadership and African-American junior officers and enlisted men.
So, African-American units faced hardships and racism from their own government (there's a story, possibly apocryphal, of a Black unit attempting to go into a restaurant to eat, only to get kicked out, while the restaurant served Nazi POWs). They also had to fight tooth and nail to receive a modicum of rights back in the U.S. after their service. Even then, they were denied many of the benefits of serving their country.
Sources:
Army Finally Recognizes WWII Black Heroes
"What Soldier's Do" by Mary Louise Roberts <---Amazing book.
FWIW, while this is already noted in another reply, the U.S. Army did not really deploy black combat units until the later stages of the war. I'm not sure of the date that occurred, but I think it would have been in late 1944. About the same time the Army actually authorized black volunteers from service units to take assignments with black combat units for that purpose, with the result being that there were quite a few volunteers who took reductions in rank in orer to serve in combat units.
This was the case to such an extent that even some very famous segregated Army units, such as the 10th Cavalry, were not deployed as combat units during the war, even though they were standing units of the Regular Army. This contrast notably with World War One in which the Army did in fact deploy black combat units to France, including some wartime units that included black officers.
The U.S. Navy, oddly, had not become segregated until the very late 19th or very early 20th Century, but during the World War Two blacks in it also served in service roles, often in shipboard service roles. The Marine Corps historically had not accepted black enlistments at all, but started to very late in the war, although again black Marines served in service roles.
So, fighting "alongside each other" really would have been an exception in World War Two. You really have to get to the Korean War before you start to see that occur, at which time the Army was desgregating.
Another dimension is the way segregated American forces interacted with the British and French, who had larger proportions of African and Indian colonial infantry fighting on the front lines. You hear stories about fights breaking out because Americans didn’t want British or French soldiers bringing an African comrade with them to a bar or something.
Did this sort of thing have any effect?