This is complicated.

There's several layers to work through.

The first is that there are a few African groups that genuinely can claim Jewish ancestry. Most notably in Ethiopia, where there is a group of people who seem to be the descendants of a Jewish kingdom called Beta Israel and who have been allowed to emigrate to Israel, though not without controversy.

For many years, a group called the Lemba living in present-day southern Africa claimed to be descendants of Jewish migrants who travelled southward through Africa. These claims were dismissed by historians until genetic testing in the 1990s seemed to verify their claims. Further evaluation of the genetic evidence has suggested that specifically Jewish descent cannot be verified, but that the Lemba do have strong genetic ties to Middle Eastern populations, unlike most of the other peoples in the region.

But as the OP says, this is only the beginning of claims that many West and East African peoples as well as African-Americans have made to have Jewish heritage. There's three different things going on here, I think, considering that these claims do not have the factual basis that the Ethiopian or Lemba claims do.

First, that there have historically been population movements as well as trade within sub-Saharan Africa that do in fact tie much of the continent to the eastern and northern Mediterranean. Languages like Berber, Hausa, Oromo, Somali, etc., are within the same language family (Afroasiatic) as Semitic languages, and a few African-based languages (most prominently Amharic) are Semitic. Northeast Africa and interior West Africa have long been connected to the eastern Mediterranean and the Red Sea world, especially but not exclusively via Islam and the movement of Arabic speakers into their world. So the history of Judaism and of the Jewish people is not unfamiliar at a deep historical level to many African societies, and in some of the major ports and cities along the East African coast and along the borders of the Sahara all the way over to the upper Niger River, it's entirely possible that small numbers of Jewish merchants were actually resident at times all the way back to the late Roman Empire.

But why might that have led some groups to identify themselves as having Jewish ties or Jewish descent in the modern era? I think you first have to see how many ruling elites in many African societies sometimes have tried to bolster their legitimacy by claiming to be from elsewhere--in interior West Africa, rulers of Mali, Songhay and connected states sometimes claimed ties to Arabs or descent from the early caliphs in the era of Muhammed; Swahili rulers; rulers of the Kilwa Sultanate in East Africa claimed to be Persian. This is pretty common in dynastic history elsewhere too: an exotic origin that ties you to events and peoples with a somewhat legendary or religiously exalted heritage is a classic way to justify holding power. And once that move is "in the playbook", so to speak, it's available to other rulers or families who want to polish their resumes or move up in status. Just think of organizations like the Daughters of the American Revolution in the US, or the predilection of many whites in the US to claim that they have a distant relationship to Cherokee princesses or British nobility.

There's still more to it, though. The other two important things that are in the mix is first a specific bit of colonial ideology that was most associated with the British Empire, something we call "the Hamitic myth". Not to go too deeply into the racist twists and turns of the concept at the height of its use, but the basic gist of it that's relevant is that British colonial officials, early anthropologists, explorers and others began in the 19th Century to argue that one of the differences between various Africa societies was whether they were truly from Africa all the way back or whether they had migrated more recently from the Middle East. And guess what? If a group were described as having some connection to the Middle East, they were regarded as "more advanced" and were often given a more privileged status within British colonial hierarchies. Some of this was a garbled and misrepresented understanding of those actual histories of connection and migration I mentioned above, and some of it was outright fiction. A particularly potent example of this was when the ruins of Great Zimbabwe were discovered by Europeans in the second half of the 19th Century, the wholly false idea that they were built by the biblical King Solomon or by other ancient Middle Eastern people was proposed (which is where H.Rider Haggard got the title and idea for King Solomon's Mines). So in this context, more than a few African groups got the message that claiming connection to the Middle East, possibly via Jewish inheritance, might be advantageous. I think you can see this partly as a sort of "humor him, play along" kind of thing until it really settled in.

But finally, the OP can likely see that some claims of Jewish inheritance are rooted in various African encounters with Christianity, on multiple levels. European missionaries frequently taught African converts a European language to read the Bible (well, Protestant missionaries did, less so Catholic ones), and once converts could read or hear someone else read the Bible, it offered a whole series of interesting and exotic narratives in both the Old and New Testament to try and incorporate into your own historical vision and your own ideas about Christian worship. For many Africans in the colonial era (and many African-Americans during and after enslavement in the Americas), the story of the Jews seemed especially moving and powerful: beloved of God and yet constantly suffering--taken into slavery, being defeated by others, building their own kingdoms under prideful and sometimes sinful kings, eventually being put under the thumb of yet another foreign empire but also being the people among whom Jesus was born. Even just metaphorically applied to African and African-American life, the Biblical history of the Jewish people offered a great deal of hope and possibility. At times, that has gone further, much as many charismatic African churches have also claimed to have direct ties of descent from Christ or to have leaders who are Christ reborn. In the case of at least one African-American religious movement, the Black Hebrew Israelites, that's gone one step further into a claim to be the real Jewish people of the Bible and other claimants to Jewish identity being pretenders.

So there's a lot of layers here to sift through when you hear any group in Africa or in the African diaspora claiming some form of literal connection to Jews and Judaism.