I think some additional reading will help you work through this, but basically think of “black” versus “non-black” as populations descended from the relatively small group that exited africa some 70,000 years ago and spread out through eurasia. (I’m simplifying, but you can read more if you like.)
This migration “gate” opens and closes during human history for various reasons, but genetically it creates a group that remained in sub-saharan africa, with descendants of that group, and a group which populated the remaining world, with descendants who populated those groups. In mtDNA terms (maternal lineage) think haplogroup M-N descendants of L3 as modern day “whites” and “asians” (again; simplifying).
Back-migration of M-N into sub-saharan africa was not robust. Further, introgression of Denisovan and Neandertal genes into non-african populations introduced several percent of archaic genes from lineages even more ancient than L0. Evolution drives changes for all genes in all populations, so modern day populations reflect the evolution of the source population genes from wherever you want to define the anchor point of anatomically modern humans (loosely 200,000 years ago) plus any changes along the way that have been driven to any significant penetrance in descendant populations. However a modern branch will not have access to evolved or introgressed genes introduced into descendant branches from anchor points which already migrated away.
For example, if MCPH1 haplogroup D variant gets introduced to M-N and its descendants, it won’t show up very broadly in sub-saharan populations because of the minimal back-migration of out of africa genes into sub-saharan africa (there were some migration gates over the years, including through the sahara from northern africa, and through east africa; but again, simplifying).
The easiest way to think of “black” as a genetic pool cluster is to think of populations which are not descended from the M-N anchor point. Another way to say it is the “sub-saharan gene pool.” Another way is “(recent) african continental origin.”
You are exactly right that there isn’t an exact genetic way to “define” a black individual or a “white” individual and so on. But you would be badly mistaken to think that the gene pool of the average individual self-identifying as black is exactly the same gene pool as the average individual self-identifying as “white” or “asian.”
That’s why we say in medical shorthand, for example, that “whites have a higher incidence of cystic fibrosis than do asians.” The gene pool from which a self-identifying white draws her genes has a higher frequency for gene variants which code for CF than does the gene pool for a self-identifying asian. And so on, for nearly every gene, since all genes evolve.
We don’t have a genetic litmus test for “asian” or “white.” We do understand that migration patterns have driven disparate frequencies in the gene pools underlying those two fairly crude self-identifications.
In other words, genes cluster differently for self-identified races, and so if we decide to use those groupings (even though there is no particular constraint to use them), the differences we see are underpinned and driven by genes.