In the old games, Carbuncle was a summon in its own right. Its abilities were mostly defensive, so things like Ruby Light that gave the party Reflect, as well as teaching things like Protect and Shell. We have a cute little nod to that in XIV with things like Aegis.
But in XIV, as I understand it, Carbuncle isn’t really its own being: it’s a magical construct formed by our aether, kind of like a familiar or a focus for our energies, which is why Carby disappears when we summon an egi: our aether is shifting to that form and calling upon that energy and the residual energies of primals we’ve been exposed to when we defeat them when we summon. So neither Carby or the primal egis are really pets, but more like magical thought forms we create. So in a way it makes sense that it’s just sitting there waiting for us to cast our next summon, even if it seems odd to how it worked before. I do feel like it could use an auto attack though when it’s not busy turning into our summons! lol