STEPS TO REPRODUCE... [1.16.3 vanilla]
Create new single player world.
Spawn in a horse, tame it and ride it [or just find one and do the same].
Log out while still on the horse and quit the game [quitting is important].
Restart the game and join the world.
You will fall off/through the horse as you load in, but you are not really off the horse, even though you can move around.
When you press the shift key you will be dismounted and standing beside the horse, even if you are a distance away; I never checked to see how far you could get though.