By no means am I discouraging new answers coming in, but I think you'd enjoy reading the relevant part of the FAQ.
This is a very difficult question to answer.
To be able to even judge whether the premise is correct ("Africa has had fewer kingdoms/less political centralization than Asia, Europe or the Middle East in recorded history") requires a great deal of knowledge. It requires a rigorous definition of what a kingdom or an empire is (is there a minimum population and territorial area required? What separates a king[dom] from a chief[dom]?) , and precise and impartial tallies of just how many kingdoms or empires there have been, and how far back are we going?
Before european colonization, there actually were a large number of centralized states in African history, probably many more than you would expect. Beyond the well known examples like Egypt, the various Nubian kingdoms, Mali, Ghana, Songhai, Ethiopia and the Swahili states, there are also less popularly known states like:
That's only a partial list, from 20 minutes brainstorming. Of course, those are all examples from historic or proto-historic time periods. That is, these states existed in eras where someone was around to write that they existed, and give some information about them before they ceased to exist.
An inconvenient fact about African history is that many societies south of the Sahara did not have systems of writing for much of their history^* . So, for huge areas of the continent we are reliant on oral traditions, linguistics, and archaeology to piece together the past. From these oral traditions we get examples like the Kingdom of Sooso or the Empire of Kitara which we don't know enough to say if they were actual entities or just stories^** .
From archaeology, it is possible to get some insight as to the level of political centralization. For instance, by looking at funerary practices, we can detect burials with lots of elite goods like ivory, gold bracelets, chinese porcelain. Archaeologists can then make some inferences about the level of political stratification and trade networks necessary to produce such grave wealth. Or, in certain regions where kingship is closely associated with blacksmithing, finding ceremonial anvils in a grave can be a strong indication that the grave belongs to a king.
The absence of writing can make doing pre-Colonial african history very challenging. I don't know of any work that has tried to rigorously and comprehensively tried to describe the political history of the continent in the way your question would require. Africa is a large continent, and the fields of African History/Archaeology/Anthropology tend to specialize by region. To create such a work would require contributions from many many scholars, and there would still be areas with limited research being done (e.g. Congo basin).
TL:DR- I don't think there is enough data or rigorous enough definitions to quantify how many kingdoms on each continent over time. With written references to lots of European, middle eastern or Asian kingdoms (and general lack of teaching about Africa), it might seem self-evident that there were more, but that is not definitively true.
*On the other hand, areas of West Africa, Nile valley, the Horn of Africa and coastal East Africa interacted with the Islamic world and either have local writing traditions or written accounts from outside travelers for several centuries before Europeans start to arrive in the 15th century.
** And of course, being written down is not a guarantee of historicity either. See for example the debate among biblical scholars over whether there was a united Kingdom of Israel before the split into Israel and Judah.