It depends on the context, and "race" is perfectly valid as a synonym for species here. See: "human race", and generally in fantasy fiction other human-equivalent peoples (elves, orcs, etc.) will also be referred to as races. "Species" sounds more scientific and sort of.... distant?
Also, cross-species breeding is rare but not impossible. Related species with similar body shapes can still potentially have offspring. (Wikipedia: Hybrid)
Of course a human can't create offspring with a cat, but different species of cats can breed with each other.
Whether the races of Eorzea should be classified as a single species or several, it's still conceivable that the different races can have children together. We might just not see many because it's culturally avoided rather than physically impossible.