I'm fairly sure of the strict RAW answer to this, but:
The Half Elf "Drow-Trained" racial trait gives proficiency in the "hand crossbow, rapier, and shortsword" - which is obviously meant to be identical to the Drow weapon familiarity racial trait. With the release of the Drow Razor, should it also allow the drow razor to be used as a martial weapon, like the Drow's weapon familiarity? Or is Drow-Trained no longer equal to the Drow's weapon familiarity trait?
It happened to be identical in function, but 'Drow Trained' never gave a half elf the "drow’s weapon familiarity racial trait" and so a Drow trained half-elf could not treat a drow razor as a martial weapon.
That being said, I'd probably allow it in my games.
My thoughts too. Seems strange an adopted human can get the Drow's full weapon familiarity but a drow-trained half-drow can't. RAW is annoying like that.