Totally agree that life is not that simple, and that people have various strengths and weaknesses.
The boxing is simply a way of summarizing those in a simplified (perhaps overly-simplified) manner.
If a developer is great at coding but a paint to communicate with and has participation issues, that honestly strikes me as a "B player" if communication and participation are key parts of their job. If not, then those factors aren't as important in terms of weighing overall performance, and that could make the same person an "A player" in a role with differing responsibilities.
I also have never worked with someone who is excellent at all facets of their job. The question is, are they excellent enough at the core aspects of the job where their weaknesses are sufficiently offset. And this is where you can compare team members. It all comes down to their responsibilities, the needs of the team/business, and how they are tracking against those.
So to summarize--everyone is indeed unique in terms of strengths and weaknesses. But at some point the weighting of those strengths and weaknesses tends to crystallize into something more easily categorized as a higher-level bucket of "exceptional", "meets expectations" (and that's not a bad thing), or "needs improvement." A/B/C is just a shorthand convention for that.