There are many shmups out there that I can't properly categorize because it lacks the option. I'm not sure how the database is set up, but typically adding such a feature is almost effortless. If you find it in your heart to add this one for me, I'd thank you.
None of them are first person, though. They are completely unlike Doom, Quake, Hexen, Soldier of Fortune, (insert every other "I can only see my gun and I shoot things" game). Look at some of these videos. SHMUP is really a different genre. Guardian Legend (NES) Gradius 3 (SNES) Radiant Silvergun (Sega Saturn) Sin & Punishment (N64) Panzer Dragoon II (Sega Saturn)
I might change the genre to Shooter then, a lot of people are confused by the FPS abbreviation anyway.
It's all in the context, I'm quite certain people won't be thinking "Hmm, I don't know any games with the genre 'Frames Per Second'
Changing to "shooter" or "FPS/Shooter" or adding a separate "shooter" would be just fine. I'm just trying to help out with the Saturn stuff since I'm a bigger Sega junkie than a Sony one and I love my Saturn.
Galaga is a 'shooter' (or shoot-em-up). That's why the First Person Shooters are seperated in the first place.