Although you are right about this being a fantasy game so such niceties as the complexties of genetics need not apply (there is after all a nice handy hint in the game's main title!), SE do pride themselves on at least trying to make the lore believable, and that includes racial crossbreeding being uncommon. On this note, I forgot to add a citation in my previous post about the playable races' ability to interbreed where Michael-Christopher Koji Fox, aka 'Fernewahles', one of the game's writers and world lore creators (and head of localization) explained about this on the old FFXIV 1.0 forum (ten years ago as it turns out!), which can be found on the 1.0 forum archive here, but I've copied and pasted the revelant info below:
We've seen this a few times throughout FFXIV with a number of crossbreed NPCs, and lets just say their lives and pasts are not pleasant due to being considered pariahs in their societies (Hilda and her hyur mother (a chambermaid) were thrown out into the street by her father, an Ishgardian elezen noble when his infidelity with her became knowledge due to Hilda's birth, and Arenvald's mother, also a hyur who was raped by a Garlean centurion, disfigured his face thinking a pimple that was growing in the centre of his forehead was a Garlean third eye (which is why he tattooed his face with a big white cross to try and hide the scar). And there is a story about a past (and long dead) Ruby Princess of Sui-no-Sato who was part au ra, part hyur who lacked scales). So it is definetely possible, but due to cultural (read: xenophobic and racist) reasons, this is a very rare event.
As I mentioned previously, there is sort of an explanation about why there are all these different 'races' that are technically all genetically similar and basically considered 'human like', but I won't spoil that, as it's part of ShB's story (and there are still a few missing aspects that just breed more questions than what they answer).