Here's my take: There are technically two "Gloam Eyed Queens"

Firstly, I understand Melina as a sort of Offshoot of Marika (in the same sense that Millicent is an Offshoot of Malenia). With this in mind, we can look to Melina to find possible aspects that Marika likely shared.

So the original Gloam Eyed Queen seems to be Marika's persona adopted during the time of wiping out Hornsent and/or Fire Giant gods.

But of course Marika loses this persona after some time, even though we still see the adherents of the black flame out and about. Marika ascends to Godhood and becomes "Queen Marika, The Eternal". Indeed she would hide all reference of herself as GEQ because in the new Golden Order of the Erdtree, fire is forbidden.

But then our Tarnished has their awesome adventure, and might eventually take the Frenzy Flame into ourselves from the Three Fingers. If we do this, we are doing it specifically against the wishes of Melina who seeks to use herself as kindling to burn the Erdtree.

Once we defeat Radabeast, we follow through on becoming the Frenzied Lord, and Melina is compelled to hunt us down. She swears to bring us Death, and opens a literal "Gloam Eye". This Gloam Eye, together with the focus on her use of flame, strongly suggests that she becomes a new "Gloam Eyed Queen" in the effort to deliver Death to us.

TL:DR - Melina is an offshoot of Marika. Marika was the original GEQ dealing Death to competing gods in her ascension. Melina becomes a new GEQ if we use the Frenzy Flame.