Since I played all the way through as Drow and never tried Half-Drow this is interesting.
Just from a lore stand point, yeah, half-drow do look kinda drowish. Like how half-elves look kinda elfish to humans and kinda human to elves. Shadowheart is a good example because Lae'zel thinks she's just an elf. But the point is the "kinda" part. So depending on what the human parent looked like you get a wide plethora of what traits they might inherit (read: what you choose to make them look like.) A human might not even think they're half-drow unless they have bright red eyes. They might see the ears, normal color eyes and pale kinda purplish skin and go "ah, mommy was a moon elf, huh?"
So it's not so much developer oversight as much as it is people look at half-elves and don't really see any subtext of what kinda elf or what kinda human mom and dad were. Elves see ears too short, humans see ears too pointy.
Drow, however, *would* notice. Because drow have been mostly selectively breeding for a few thousand years. They see someone with enough drow traits and go, "Ah. Your mother had a human concubine then?" at best and think you're the drow equivalent of a mudblood at worst.
So your character *should* make half-drow remarks, and probably half-elf remarks, but a big chunk might see a half-drow and not realize the drow part.
In reality, maybe more people might jump and think drow, but that depends on region of Faerun and the person running the game. Remember: in the game world there are a lot less half-drow population than drow. (Your character is unique and doesn't equate to in game population ratios, because in the pen and paper version you can play as a bird man that has power armor if you really want to.) So, maybe Larian should make or has made in the areas we haven't seen more half-drow or people seeing a half-drow and thinking "oh, drow!" reactions.
