Did some testing on this for a few days and as far as my account goes, I learned if I login to a toon, or port to a new instance with a toon and then my first form of movement is sprinting on foot, that toon will get the stamina bug...which effects not only mount, but toon stamina (YES, toon stamina as well)...If I move in any other way, the bug does not appear. Of note, I did not testing jumping or dodging as my first form of movement.
Again, this goes for porting to another instance as well, so if you do not have the bug in one location say you're in Rimmen, then you port to Solitude and sprint on foot first thing, you will incur the bug.
This testing was with two of my toons only, but worked exactly the same for both. Obviously more testing over multiple accounts would need to be done to narrow this down more.
The workaround is simple, your first movement after logging into a toon or porting to a new instance CANNOT be sprinting on foot.
With that said, after seeing the 6.1.6 patch was released on September 7th and this bug was not addressed, I have decided to stop my own testing and go with what I have learned as it is apparent Zenimax only cares about what they care about and nothing of their customers concerns. Good luck to rest of you with this bug...try my work around, it might work for you as well.
Take care,