Crudely speaking, tieflings are half-human, half-devil. However, as the answer to the other question (the one about tieflings) points out, non-human half-devils/demons exist, such as the Fey'ri, who are half-elven, half-demon.
Aasimar are half-human, half-angel (again, crudely speaking; it's more like mostly human and a tiny fraction part-angel, as I understand it). Taken from Volo's Guide to Monsters, pg. 104, here are some quotes that explain why I believe them to be so:
They are descended from humans with a touch of the power of Mount Celestia, the divine realm of many lawful good deities. Aasimar are born to serve as champions of the gods, their births hailed as blessed events. They are a people of otherworldly visages, with luminous features that reveal their celestial heritage.
Despite its celestial origin, an aasimar is mortal and possesses free will.
Hence I conclude that they are part angel, even if it's not necessarily down to genetics (that is not what this question is about anyway). This question is about the human part, and whether that must always be the case.
Are there any non-human half-angels races, such as a half-elven, half-angel race?
I'm not that interested in a specific setting, but if setting matters, let's go with the Forgotten Realms (or at the very least, not Eberron/Ravnica).