The first time it happened, you didn't know enough to notice.
To be fair, the giant angry fire monster was a rather imposing distraction, and you had more important things to worry about than why, exactly, you were the only captive to keep their mind.
Your memory is a blur of fire and desperation, K'lyhia's lessons echoing in your mind as you shredded ruinous corrosion through Ifrit's body and nails. In the end, he exploded into aether, as primals are wont to do; occupied with healing the wounds rent open when Glittering Topaz shattered under the onslaught, you paid no mind to the motes of light which sank into your skin, melded invisibly with the air and slipped down into your lungs.
You were alive, and untempered, and desperately needed a nap. Dizzy with exhaustion and the sharp euphoria of survival, the echo of flame smouldering inside you went forgotten until Y'mhitra called you to drag him back out.
It was a strange feeling. Your mind was still your own - you would have been executed if it wasn't - but a section of your essence was entwined with the memory of burning wrath, a black sun above and blazing stone beneath. That first manifestation was built from a core of searing embers, scooped out from your body and set aflame by the light of the desert.
(It felt a little bit like that time Yda dared you to eat a dragon pepper raw.)
The Austerities stoked those embers into an explosion, the heat bursting out from between your ribs and blossoming into a creature of pure, undiluted flame. The egi did not burn unless you wished him to; the same claws that carried stacks of books with ease would sear flesh down to the bone on contact.
When you looked into yourself in the aftermath, fumbling through the rites of introspection gleaned from ancient literature, the initial glowing coals had flared up into a brilliant warmth. Ifrit's fire had made a home for itself in the depths of your gut, flickering with anticipation as you felt along the edges of its essence.
Summoning gave you a power unlike any you'd ever known, arcane or mundane. Your egi's strength was heady, his flames licking up your spine, and when Y'mhitra mentioned Titan you were already nodding along.
In arcanima, one of the oldest and most vehement debates concerns the sentience - or lack thereof - of carbuncles. Some scholars argue that they are no more than extensions of the arcanist's will, anthropomorphised by mortal minds, while others give their pets names and treat them with the care afforded any beloved pet.
The situation at hand is both identical and different. Your egi was created from the pieces of you infused with Ifrit's power; he is, very literally, a part of you. But he has a distinct will, independent of your own. He loves to train and bask in the sun and refuses to go near chocobos after yours tried to bite him.
You named him Blast, in the end, after what you felt when he came into being. It comes in handy, especially in situations where referring to a friendly creature using the name of a monstrous primal would cause significant unrest, and as such is entirely justified.