To reproduce:
Spawn a dolphin
Give yourself some armor
Throw them at the dolphin
→ See the dolphin pick up the armor, but not drop it again like any other item.
Notes:
This applies to other armor too.
The bug is also shown in the nbt of the dolphin in ArmorItems, when you run the command /data get entity @eArmor[limit=1,type=dolphin] ArmorItems
Confirmed for 18w19a. Although it seems like Depth Strider no longer gives them super speed.