Scientifically speaking, there's no such thing as race. There's species and subspecies. Beyond that, there are ethnic groups and there's geographical variation. All modern humans (i.e. everyone alive today) are the same species and subspecies,
Homo sapiens sapiens. This is based on the amount of genetic divergence, i.e. way less than people think. We're all descended from a very recent African population, probably from around 60,000 years ago. Compared to Neandertals for example, who are usually classified as a different species - their lineage and ours split somewhere around 600,000 - 1 million years ago. More than ten times the amount of genetic difference.
For fantasy peoples, if you're going to be technical, they should be called different species, unless they are actually closely enough related to us to be classified as
Homo sapiens. As there's no scientific definition of race, in this context it's fairly meaningless. If you want a better term then I'd suggest different peoples. I am vehemently against the notion that only
Homo sapiens sapiens people can be called people or human. Scientifically, "human" is any species in the genus
Homo. Hominin is any species in the human clade/lineage since we split from the chimpanzee/bonobo clade/lineage.
I would use "person" for any species that has a similar level of intelligence to a human. I also support "
great ape personhood" legislation, so therefore wouldn't just use "people" for species that have exactly equal intelligence to humans. So for various fantasy and alien peoples, this word seems the best choice for me.
I've taken to using "people" in this way a lot, because species names for earlier human species sound too coldly scientific, like you said in your post, plus most species names are a pain in the arse to pluralise* (there's no plural, you have one
Homo sapiens, one
Homo erectus or you have lots of
Homo sapiens, lots of
Homo erectus) but writing about a
Homo erectus person or
Homo erectus people, for example, not only gets over the grammar issue but also recognises their personhood. One of my current WIPs, the MC is a cloned Neandertal in a contemporary setting (well, 2050s, that's nearly contemporary...) and have the problem that there's no word for modern humans besides
Homo sapiens sapiens, anatomically modern humans (AMHs), etc. Even Cro-Magnons won't do because that's a name for a population found in Europe (i.e. the first
Homo sapiens to move into Europe from Africa), so I've taken to referring to modern humans as
Homo sapiens people. My MC considers himself to be a different species of human and a Neandertal (spelled the modern way without the H, and frequently annoyed by anyone who spells it with an H).
*please don't use "Homo sapien" (or worse, homosapien) as a singular for Homo sapiens... *shudders* ...don't even go there! It's like wearing a badge that says "I know nothing! I can't even spell my own name!"
Also note that while Neandertal is the correct modern spelling of the place in Germany, an up-to-date spelling of the name for the human species, the Latin name retains the H, i.e.
Homo neanderthalensis. The pronunciation has always been "Neander-tal" and "neander-tal-ensis" for the English and Latin names respectively.