I'm going to look at the claim as posed: that black people were the first people to own slaves, presented as a definitive claim: no qualifications, caveats, or restrictions (such as black people were the first to own slaves in such and such a time or place...). Spoiler alert: as you guessed, it is wrong to make the claim as posed. The origins of slavery are prehistoric; we do not know where and when it first emerged, who were the first enslavers, or who the first enslaved; prehistoric slavery may well have independently developed more than once.
From the way you state your questions, this is how you took the claim, which is why it's what I will eventually get around to focusing on, but I would also bet that your friend intended to make a more narrow claim about slavery in Africa pre-existing the beginning of Atlantic slavery, rather than the impossibly broad claim she actually threw out that reads as a sweeping assertion about the entirety of slavery. It's pretty common to see discussions of early modern enslavement of black people in the Americas (in academic writing, often referred to as Atlantic slavery or the Atlantic slave system) get interrupted by "but what about preexisting slavery in Africa?" Perhaps your friend has heard such talking points and intended to reuse them but did not remember specifics.
Briefly, then, before I do look at the claim as posed:
- It would be true to observe that slavery existed in Africa before Europeans began setting up the Atlantic slave trade to exploit African slaves in the Americas. Systems of slavery in Africa deserve and receive study, including study of their interactions with the Atlantic slave system.
- That doesn't mean the existence of the one should be used to derail and shut down discussions of the other, as often seems to be the case (but detailed discussion of current politics would fall under the twenty-year rule, besides the fact that modernity is not at all my wheelhouse).
- There are great numbers of previous answers that cover slavery in Africa and the interactions of slavery in Africa with Atlantic slavery and other slaveries beyond Africa itself; some also touch on the topic of whataboutism w/r/t Atlantic slavery and the others. A few of these: this thread with answers by u/swarthmoreburke and u/tropical_chancer as well as followup discussion by u/Commustar that includes links to several more answers about slavery and Africa; this answer by u/sunagainstgold, reviewing perceptions in Africa of indigenous slavery as opposed to Atlantic slavery at a time when both systems operated; this one with commentary on slavery in West Africa before the growth of Atlantic slavery by u/Halfacupoftea.
Okay, so. Some comments on various aspects of...
The Claim As Posed
- What the written historical record can say about the origins of slavery as an overall human institution is that those origins predate the beginnings of the record itself. Slavery is not presented as a novel innovation in the very oldest written records of it preserved from the first societies to develop writing circa 5,000 years ago, but rather seems to be an already well established social institution. This tells us that the beginnings of slavery are literally prehistoric (and plausibly not just prehistoric by a little while but prehistoric by a lot, on which see below).
- Therefore, the question to some extent passes out of the province of historians and into the province of anthropological theory and modeling - particularly drawing on archaeology and on inferred comparisons between unattested aspects of long past societies and attested aspects of more recent ones. For real depth about anthropological theories relating to the prehistoric origins of slavery, you may want to try to get an answer over at r/AskAnthropology.
- Briefly reaching for some amateur anthropology, though: within historical times (i.e., using the comparative approach), the institution of slavery is well attested within societies practicing neither settled agriculture nor urbanization, suggesting that neither is some kind of hard social prerequisite for the development of slavery or of social institutions that are more or less similar to slavery. This suggestion does nothing to prove that slavery or slaverylike circumstances in fact are older than the Neolithic revolution and the spread of settled agriculture, it just means that we would not on those grounds rule out institutions like slavery predating even the Neolithic. If you do ask over on r/AskAnthropology, though, you may get some discussion about the social implications of Neolithic developments such as settled agriculture, early urbanism, increasing appearance of archaeologically visible wealth disparities, and the more stratified social hierarchies that are often theorized as having codeveloped with those disparities. Perhaps a discussion about whether these and other Neolithic developments that are archaeologically visible or directly inferable can be interpreted as catalyzing and/or intensifying the development of slavery and similar social institutions that are not themselves archaeologically visible and are only inferable more indirectly, if at all.
- By the times the historical record does pick up (which are different times in different places), it shows slavery already widespread throughout the world. Slavery and slaverylike institutions existed, for example, in the precontact Americas. This suggests that the migrating ancestors of Native Americans brought the institutions with them across the Bering Strait (making slavery very very old indeed, old enough to have existed for a pretty notable fraction of the total span of existence of behaviorally modern humans, which is waaaay longer than "black people" have existed as any sort of broad social grouping) or that they developed them independently, meaning that institutions of slavery emerged in multiple times and places rather than all tracing their descent back to a single time and place in which slavery was invented for the only time in human history. Or both of the above, but at least one of them. Personally, regardless of the unanswerable question of whether slavery or something close to it already existed in the peoples who crossed the Bering land bridge, I think* it is probably more plausible to suppose that prehistoric slavery emerged in multiple times and places more or less independently than it is to suppose that it only ever developed in one time and place and then spread throughout humanity from that single origin.
Putting this together into something resembling an answer to part of your question, the identity of the first humans to ever enslave others is irretrievably unknowable, and therefore the precise, or even ballpark, pigmentation of their skin is of course also unknowable. It is certainly wrong to claim to definitively identify them with one particular modern category of people, such as black people.
There is also the issue of whether there ever was any single group of original enslavers in a single time and place at all (their identities prehistoric and unknown) rather than multiple more or less parallel but independent origins among various groups in various times and places (their identities mostly likewise prehistoric and unknown).