Yes.
But only because it's the first game worth its own genre.
Now, if I back off that claim a bit, I'd have to say, 'no, it's not my first horror game.'
The first one might be
Moria. How can a DOS screen of static ascii characters be horror? I don't know. Something about going down, down, down into the depths is frightening. You didn't know if at any given moment you would be put in a perilous situation. You felt attached to the protagonist.
Ok, it's a stretch. Maybe the first horror game was the original Castlevania, or, more likely, Doom. I was a college freshman when Doom rolled into town, and that was my introduction to multi-player gaming. I played a ton. Doom I, Doom II, and Quake were all hits while I was in college. Because I lived in dorms with everyone else at a military academy, it was a popular way to kill time. Not much of those games really resembles Amnesia, but I know those games led to nightmares. The disturbing and psychological environment was the only real claim to horror I suppose.
During my junior or senior year I think the first Silent Hill or Resident Evil came out. Suitably scary, and probably my first 'true' horror game. I didn't play any other horror titles until Doom 3 (edit: does Diablo2 count?), which I thought was a good game to play at midnight in a dark room with surround sound. It was scary, but looking back at it with my Frictional experience, the ability to shoot monsters results in a warfare game, not a horror game. Right before I played Amnesia, I had played Bioshock, but I didn't find it remotely as scary, except in a few parts.
So, yeah, even though I'm 36 and I've played one or two games, Amnesia is the first horror game. But, to be fair, I haven't played them all--I missed FEAR, System Shock, and the Cthulhu game. Are they worth going back for a visit, or will I be disappointed post-Frictional.
Ah, Frictional, you've ruined me for other men... lol