I am not able to reconcile what D&D really means by "race", other than the tables of abilities in the player's handbook.

The basic biological taxonomy goes as follows:
kingdom,
phylum,
class,
order,
family,
genus, and
species

Then where do Tieflings branch away from humans? They seem to have a spinal cord, so the phylum "chordata" is the same. I would guess they could be considered mammals? But on the other hand, maybe they could actually be in a different class. Once again, the virtue of unknowing reveals itself.