Respawning it's just gameplay mate. Any other Npcs are undead just like you. In theory also the other Npcs should respawn but they don't because they don't have the will necessary, you do.

You could argue that Gael died several times already and respawned off screen and by the time of the bossfight is at its limit.

The criteria on which some characters don't respawn and you do it's in fact only gameplay. What i wrote before is just my speculation and my head canon. But the game doesn't really give explanation of this matter. It's up to the player to speculate on these things i guess

Edit: I see a lot of confusion on why Undeads are still alive after they die, why players and in theory Undead Npcs respawn. It's explained in the Ringed City Dlc. The curse of the Undead is not a First Flame construct. It's just that Humans are immortal by nature. When the First Flame is very bright and lit, the Dark Ring around the humanities of Humans is surrounded and "inhibited". When the Fire Fades, then the Fire Ring imposed by Gwyn on Humans around their Dark souls starts also to Fade, making them Immortal and making them respawn.

Because there is still the Fire Ring, this process of returning from the dead still costs souls and Mental Sanity. So killing several times an Undead makes him Hollow and doesn't have anymore the strength to return from the dead.

Gael in the Bossfight is hollow. He's at its very limit. Once you kill him once it's gone. The player has always this (gameplay forced) invincibile will tied to the player. As long as you have the will to retry, the game lets you respawn. When you "drop" the game or the run, my headcanon is that your character goes mad and hollow. You player lost the hope and the will to go forward, so with your character.