The basis of my own thought is simply the existence of the soul. In D&D, the soul is very real. You can cast Speak with Dead and converse with the soul. It's shepherded to an underworld that you could actually visit, by a goddess that you could actually converse with. You could summon the soul back to bring the person back to life, or to bind to an undead shell.
The intelligent species all have souls (as far as I am aware), and they were all created by gods. Any that don't have listed creators are still assumed to have been created at some point, somehow, by some god. Non-intelligent species don't have souls. (Yes, elves didn't have souls to start with, but they have for a long while now, so I'm mostly ignoring that quirk of history.) So my interpretation is that souls are what make intelligence creatures what they are. They are what the gods created, and what then gave form to the physical body of each race.
A new soul is created from the intermingling of the souls of its parents, like ink mixing together. There's a bit of influence from the individuality of the parents, plus a bit of chaos mixed in, but the core remains itself. The child of human parents is human because the mingling of the human souls created another human soul. On the other hand, if the parents were human and elf, you'd have a typical half-elf.
And, particularly, the body is shaped by the soul. A human soul will grow a human body. A dwarf soul will grow a dwarf body. A half-elf soul will grow a half-elf body. Etc. There is no restriction on breeding due to species or genus or whatever, because it is not genetics which defines the child's form.
I would speculate further that for elves, beyond just the mixing of the souls, the child is influenced by the spirit of the land around it as well. For example, elves that live in or around the sea would take in certain aspects of the sea when their soul is created, allowing breathing underwater. This leads to the "speciation" that gives us sea elves, or wood elves, or dark elves, or whatever. (This can conceptually tie in to the older idea of elves having "spirits" rather than souls.)
Overall, two parents having a child create a child soul which binds to a physical element (the mother's egg) that grows and is shaped by the essence of the soul itself. The physical body will grow to match the soul's identity.
So, any god-created race can interbreed with any other without having to jump through hoops trying to explain things using genetics or other real-world stuff. Species is not a thing, and doesn't matter.