As far as I know, the "fish count" achievements only count items that a Hunter's pet would consider "fish". Broadly, anything that can be cooked. Alchemy fish, trunks, junk, and similar do not count. If you fish a lot of Azeroth's pools, it is possible to make no progress towards this achievement at all.
That's what I thought at first as well. Alchemy ingredients not counting as fish was the intuitive answer, and I thought I specifically checked for that possibility. Apparently, my eyes were either deceiving me or there was some kind of display lag at play, for you seem to be right now that I've checked again while fishing in a different area.
I'm fishing for the Crushfish on Menethil Harbor (surprisingly easy to fish there as a Horde without grabbing guard aggro), and it appears that my Firefin Snapper and Oily Blackmouth do not count. I don't think the hunter pet litmus test is really a good one, though. I got into fishing to feed my kitty, myself. Hunter pets that eat fish will eat almost anything remotely fishlike. They even eat the toxic "Do Not Eat" fish from fel pools. I just tried to feed both Oily Blackmouth and Firefin Snapper to my cat, and in both cases, I got the "That food's level is not high enough for your pet." message. If it's something your pet never would have eaten, you instead get, "Your pet doesn't like that food."
I think it's just an oversight, that they're excluding the alchemy fish because they're considered part of a different profession. Or they might be excluding inedible fish. I'm going to try this in some different areas later. I'm curious as to how it will perform with "questionable" fish of other kinds.