For the achievement “No Stable Big Enough”, I have a count of 345/350 on all of my 8 characters. I gained two new mounts this last week (Xinlao and Drake of the South Wind) and the count has not changed. My pally, Foehammerii (Echo Isles) has the most USABLE mounts at 347 (according to the Armory and by actual count in-game). This happened last month when my achievement count was 341 and I got two new mounts and no increase in the achievement count until I actually learned a new mount on my pally, then the count was updated to the proper number. The last two mounts I got were obtained on characters other than my pally. I have logged in on ALL my toons to check the achievement (no counts were updated), have also checked all toons on the Armory (counts for the achievement range from 343 to 347). One person said it could just be a UI error, /reload ui on each toon did nothing. The Armory has the correct counts for each of my characters (all on Echo Isles), Kwai-343, Foeshadow, Tattertot, Ringmule, Glormara, Grayhame all with 345, Foecleaver (my DK) with 346, and my pally Foehammerii with 347. The achievement SHOULD be tracking and reporting my pally’s count at 347 but has not updated.
If you feel that it’s bugged, you’ll want to post in the bug report forums. That said, is it possible that the last two mounts you added are not ones your pally can use? For example, are they class mounts or faction specific mounts that wouldn’t count towards their total?
This one isn’t faction- or class-specific
Have you tried resetting your UI using Resetting the WoW User Interface - Blizzard Support and not the /reload command. There could be an add-on blocking the update in-game. Renaming or temporarily moving the folders will allow you to restore them and your settings after you test.
Ignore the counts on the Armory, Achievement, mount page and most mount-tracking websites.
Hit “Y” and go to your achievement page. Click on the Statistics and go to Character > Gear > Mounts Owned. That’s the only count that matters and it’s always up-to-date.
The last two mounts were Xinlao and the Drake of the South Wind, both usable by any/all characters.
Stats say 347 mounts owned, achievement only says 345.
I don’t use ANY add-ons. Will try the link you posted for resetting the UI.
Sounds right. Ironically that’s the count your achievement uses. It’s been screwed up for along time. Just keep checking that statistic and ignore everything else.
I still think the correct count will only update when the character being tracked for the achievement (my pally in this case) actually learns a new mount himself, so I’m running Stonecore until I get the mount from Slabhide. 
Think that if you want but I’ve been collecting for along time (I’m at 518 mounts on my Paladin) and the statistics page is the only place to get an accurate count.
Good luck.
It should update on login, though I’ve had it take a few login of different characters to see the change.
Have you tried using the mounts on you Pally?
i had an issue with my last mile-stone.
i reported it, but… yeeeeeah. 
here is what happened with me 
That’s what happened to me last time, the achievement count was at 341. I got the mounts from Ren the Houndlord and Ishak (one on my hunter, other on my pally). Count didn’t update when my hunter learned the mount he got, but did when my pally learned the one he got.
Just tested the theory. Bought a dune scavenger from the AH on my mage and sent it to my pally. He learned it and the achievement count went from 345 to 348 (it should have been 347 before the mount, 348 after is right).
So Blizzard, the game IS NOT keeping track of mounts correctly. Get to work on it!!!
If you want them to work on what you think is a bug, it needs to be reported in the cleverly-named Bug Report Forum.
Posting here will not get anything done.
OP already placed a bug report, a bit before coming here. I wasn’t able to help much; some folks here had some information that I didn’t know about, so that was awesome. But definitely a bug, and already reported by the OP 
For those struggling to understand why sometimes the achievement does not increase, after some testing I came up with these findings:
Let’s start with a simple scenario where on your account you have a rogue and a warrior and both have 10 mounts. No unique or exclusive mounts present, both characters can use 10 mounts.
You can check on Statistics > Gear > Mounts owned that both characters have 10 mounts.
You can check on the mounts Achievement that you have 10 mounts.
Now let’s pretend you got a gladiator mount on your rogue, which currently is exclusive to the rogue.
At this precise moment, the rogue mounts owned will go to 11.
The warrior mounts owned stays 10.
A script will run for your Mount achievement progress and check, is the mounts owned of this character (the rogue) greater than the current progress? Yes, because 11 > 10, so update the achievement progress also to 11.
So current status is
Rogue mounts owned: 11
Warrior mounts owned: 10
Achievement progress: 11
Now let’s pretend you got a mount on your warrior, but not an exclusive one.
At this precise moment, the warrior mount goes to 11.
The rogue mounts goes to 12.
A script will run for your Mount achievement progress and check, is the mounts owned of this character (the warrior) greater than the current progress? No, because 11 is not > 11, so DO NOT update the achievement progress.
So current status is
Rogue mounts owned: 12
Warrior mounts owned: 11
Achievement progress: 11
THAT’s why sometimes you will get a mount but not see an increase on the achievement tracker!
Now let’s pretend you got a mount on your rogue, but not an exclusive one.
At this precise moment, the rogue mount goes to 13.
The warrior mounts goes to 12.
A script will run for your Mount achievement progress and check, is the mounts owned of this character (the rogue) greater than the current progress? Yes, because 13 is > 11, so DO update the achievement progress to 13.
THAT’s why you sometimes will get ONE mount but see an increase of 2 on the achievement tracker!
So if you want to “fix” your achievement progress, you just need to get a mount on any character where their mounts owned is bigger than the current achievement progress.
While Blizzard support article (https://eu.battle.net/support/en/article/4738) do say that you can log on other characters to trigger an update, I didn’t experience that working reliably.
Only by getting a mount on a given character where their mounts owned was bigger than the achievement progress that I saw the achievement being updated all the time.