When you enter your home instance, the game should dismount you. Then the mounts would roam around the home instance like cats current do (or the natural raptors in the story mission). This way you can:
see your mount as a NPC (name your mount, talk to it)
play with the mounts
buy pet related items for the mounts
bring gifts for the mount like we did for Aurene
do things to get 24-hr mount-related buff like we do with Guild tavern buffs