The noun, 'Gigi' is a proper name.
The name 'Gigi' may be a nickname for someone, but the proper name for that person can be anything at all, including their initials (G.G.), there is no standard for the nickname Gigi.
Famous Gigi's include:
We also refer to my wife's grandmother as Gigi (as short for Great-Grandmother).
Gigi Rua's birth name is Gigi Vila Rua.
Gigi de Pourtales's birth name is Marguerite Lois de Pourtales.
Gigi (singer) birth name is Ejigayehu Shibabaw.
Gigi Sumpter's birth name is Jennifer Sumpter.
Gigi Reder's birth name is Luigi Schroeder.
Gigi Ballista's birth name is Luigi Ballista.
Gigi Proietti's birth name is Luigi Proietti.
Gigi Oeri's birth name is Gisela Trefzer.
Gigi Orsillo's birth name is Jeneva Graham.
Gigi Gatti's birth name is Fiorella Gatti.
Gigi Levangie's birth name is Levangie, Georgianne.
Gigi Gore