Actually FF's Gilgamesh (as with most of their 'summons': Ifrit, Shiva, Quetzalcoatl, etc.) is based somewhat on tales from mythology
https://en.wikipedia.org/wiki/Epic_of_Gilgamesh
His story is split in 3 parts:
The demi-god tyrant of a king who after intense battle befriends the man (Enkidu) sent by the gods to kill him.
The mighty warrior who fought terrible beasts and monsters alongside this new friend.
The wanderer who donned a life in the wilds whilst mourning the death of his friend and (possibly in fear of his own inevitable end) goes on to seek the secret to immortality.
You could base his skills off of these original tales, so something that implies his royal/divine nature tinged with ferocity and bloodlust.
Possibly some form of self sustain (healing, regen) or self buff(s) for his royal/divine side,
and heavy multi-hitting attacks (maybe with recoil) for his warrior/wild side.
He could even fight in cycles: passive/defensive stance with buffing/healing/cleansing followed by an onslaught of vicious attacks then back to his defensive stance
Interaction with this new world:
Feral beasts would likely remind him of his days in battle alongside Enkidu, possibly sending him on a rampage as he remembers the death of his greatest friend.
But it's possible one of the cuter critters could remind him of the good times calming his rage, to the point that he names it after his long-lost friend.
I can just imagine this beefy ferocious warrior with a cute hamster on his shoulder