Before, every time I learned a new mount, I would load each of my characters and the count would update properly, but that has stopped working.
I found this post: Mount count not changing since 8.2.5 - #4 by Aracina-windrunner which describes my problem perfectly, so I am not alone and this has been an on-going issue for a lot of people. Please look into this.
Edit: just to clarify in case you don’t want to read through the post I linked - the problem is that the count will not update until the character with the highest count has learned a new mount. As I pointed out, before you could just log in to that character and it would update (as annoying as it was). Now not even that works.