I cannot speak for other Ancient Civilizations but, at least for the Romans, yes they did. A lot of the sources we have come from ancient graffiti at Pompeii and other well preserved archaeological sites. There are many things carved into the walls, among them were ads for prostitutes and even reviews. Some of the names we have found are regular Roman names, but others have a distinctly prostitute sense to them. Consider the name Culibonia: it is a corruption of common Roman matron names (Scribonia etc.) and culus [anus]. A prostitute with this name would have probably specialized in anal sex. Other names might advertise their ethnicity, such as Attia of Greek origin.

For further reading I would suggest:

1. The Economy of Prostitution in the Ancient World by Thomas McGinn which has a list of prostitutes (which I believe is online somewhere but I can't find it at the moment) and includes their names as well as other information like gender and what they charged.

2. The Latin Sexual Vocabulary by J. N. Adams This is a more advanced text and requires a decent amount of knowledge in Latin and a little Greek but would be a helpful resource to consider if you want to look into names like this more in depth and understand the names of prostitutes better.