Two things.
A) Different universes, different rules.
In FF14's universe, Carbuncles are aetheric constructs created by Arcanists through the use of arcane geometries (more specifically, gemstone-based ones).
They're not Primals.
At best, they serve as the "focus" for various ACN/SMN skills.
B) There are various other summons from past games that didn't become Primals in FF14 (thus, won't have an Egi/Trance/etc. made out of).
Such as Kujata, Siren, Diabolos, Mist Dragon, Eden, etc.
So yeah...