Steps to reproduce
Make a 1x1 enclosure with a dispenser facing into it on one side.
Spawn a skeleton.
Confirm that the skeleton cannot pick up items by dropping a sword on it. If it picks up and equips the sword, kill it and go back to step 2.
Pick up the sword.
Disarm the skeleton using
/replaceitem entity @e[type = skeleton] slot.weapon.mainhand 0 airPut a bow in the dispenser.
Activate the dispenser with a redstone signal.
Expected result
The dispenser equips the skeleton with the bow.
Observed result
The dispenser shoots out the bow and it floats on the ground.
Unarmed Skeleton (In Creative Mode) can pick up swords and even sometimes axes, but they DO NOT pick up any given bow, either enchanted or not. It doesn't matter if the bow is dispensed or dropped.
Not all monsters like skeletons and zombies have the ability to pick up items. They have a chance of spawning with that ability based on difficulty level.
When I tested disarming skeletons and dropping bows, I found that unarmed skeletons that had the ability to pick up items do pick up bows. I confirmed they had the ability to pick up items by dropping swords for them before disarming them. This is expected behavior and not a bug.
However, there does seem to be a bug with dispensing bows. Dispensers should equip items to mobs regardless of whether the mob has the ability to pick up items on their own. I found that this works for swords but not for bows. Skeletons that will not pick up swords can be equipped with swords by a dispenser, but they cannot be equipped with bows by a dispenser.