Black people in ancient Rome were not discriminated against because of their skin color or physical features. They were not excluded from any profession, and there was no stigma attached to mixed race relationships. Classical writers did not associate social status or degree of humanity with skin color.
And this is why I tend to get frosty when hate groups try to use the Roman Empire to justify and/or symbolize their hatemongering. The city itself was the largest melting pot in the world at the time, surpassing even Alexandria in both size and diversity. I don’t know if there are any official numbers, but I’d be very surprised if Rome wasn’t in the top five most culturally and ethnically diverse nations that ever existed.
In the ancient world people didn’t really discriminate based on race (not to say they didn’t have stereotypes about people – Greeks were often sought after as teachers by Romans because they were thought of as intellectual), but on nationality – The world was just divided into Romans and everyone else. Although even someone who wasn’t a Roman citizen, like a freed slave, could rise to become quite prominent in Roman society.
The Greeks were much the same – they had one rule which was that you couldn’t enslave fellow Greeks (which the Spartans did, and no one did anything about it because they were the Spartans).