I understand that there is a bunch of help on this issue already with it only counting what can be used and only on a single character it tracks. I understand that and that is why I am putting this is the Bug Report Forum. I have been sitting at 343/350 on No Stable Big Enough for what seems like months. That count has been on my Druid and Warlock and only those 2 do I run mount runs and collect mounts. I have gotten several mounts between getting 343 and now, such as the Epic Edition Shadowlands Mounts, 15th Anniversary, Headless Horseman’s, The Mechagon Mechaspider Mount and the Level 20 Neri Snapdragon. All of which are Horde useable mounts and have been collected on my druid. However my count has yet to go up to 344,345, etc. Now I know its a single character but I got Invincible on my Rogue and that counted on my Druid in Legion. I got all the argent tournament mounts on my warlock and I got Kuafon not long after on my druid and my mount count increased. So to me this seems to be a bug in mounts not counting. Sorry for the long post but I wanted to give as much detail as possible to help identify the issue.
I know it can be confusing, but you need to think of the mount count shown on the achievement as a “bar” rather than a count of total or applicable mounts on the character you are currently playing.
Let’s say there was an achievement for owning 10 mounts. You play something like a druid and you collect five mounts. Achievement now says 5/10.
Now, you log in to a warlock, which has (at least) two more mounts than a druid from their Vanilla class mounts. This will increase the total shown on the achievement to 7/10.
Now, you log back into your druid. The druid only has five mounts towards the achievement, like we said. But, because the warlock raised “the bar” to 7, the achievement will still show 7/10 on the druid.
When the druid (at 5 mounts) earns his sixth mount, the achievement will still say 7/10. When the druid earns his seventh mount, the achievement will still say 7/10. It wouldn’t be until the druid earned his eighth mount that the achievement would increase to display 8/10.
Now, extrapolate this out to an entire collection that has hundreds of mounts in it. Toons like warlocks can have five or more mounts that druids don’t have (2 vanilla class, 3 legion class.) Ditto DKs with their class mounts.
You may find yourself in situations where obtaining several mounts may not increase the total. The best way to see the actual total is to log into a character that has a bunch of class mounts. Warlock. Paladin. Etc. I know it’s confusing at first, but not so much after you think about “the bar.”
Please check my answer here Achievement not updating mount count. I also give a similar explanation to the one above.