Dragon Souls are entering my mount's inventory when dragon is killed. Water Dragon does not seem to drop one though.

You can gain souls by: Spoiling the souls (spoil via inventory or use it as fuel for your Dragon Forge , eating Dragon Egg (water, toxic, lightning, fire, frost and earth), killing them, salvage a Dragonplate Armor in a grinder.

Probably look in a dinos inventory or your's when you killed it or maybe look on their corpse / bag if they have it

See if there's a corpse bag there. It happened to me and I found it in the corpse bag. Apparently sometimes it drops right in inventory and sometimes it drops in a bag.