I took the saddle off a horse.
A few seconds later it reappeared on the horse.
I was able to Ride and control the horse.
Opened the horse inventory:

• The saddle slot was empty.
• The horse model showed the saddle being present on the horse.

Possibly relevant that I was getting serious lag at the time.

UPDATE:
I've updated the title to be more accurate, this actually happens when you leash a horse after removing it's saddle. I can recreate this reliably as follows.

• Ride and saddle a horse.
• remove the saddle
• dismount
• leash the horse.
• A duplicate saddle appears on the horse but the removed one remains in your inventory.
• remount the horse
• Open horse inventory
• Observe the saddle slot is empty but the horse model on right shows the saddle present as per screenshot
• the horse can now be ridden and controlled as if it was saddled, without costing you a saddle from inventory

Can anyone else confirm this?