That phrase is rather imprecise. What can be attributed to the Sumerian cities is the first appearance of several professions which are not directly productive

That phrase is rather imprecise. What can be attributed to the Sumerian cities is the first appearance of several professions which are not directly productive and which have remained important until today. By profession, I mean an activity that was all that a certain human did in order to obtain all the resources required for living. Before Sumer, those activities were done by certain people, but only occasionally, besides their main craft, e.g. a shaman was likely to also act as a medical doctor when needed and a chief of tribe was likely to also judge conflicts. Among the professions that have first appeared in Sumer during the third millennium BC, are: judge, medical doctor, accountant, musician, prostitute. The equivalent of priests and kings, though only at smaller scales, e.g. for a tribe or a confederation of tribes, have certainly existed much earlier than Sumer and Egypt. "Chief" is likely to be the oldest profession, as that exists at many social animals, including apes.