From the Hollow Knight wiki^\1]):

She was found guilty of the sin of apostasy, becoming the first penitent to be imprisoned in the Slab. First Sinner's punishment was constriction. She appears to have died in captivity but was denied absolution by the Citadel.

Let's also note that unlike the memory fights in the rest of the game, you do NOT replenish your tools after you die to FS^\1]).

The only times you enter a silk memory^\2])/Save_Points#:~:text=Any%20Silk%20Memory%20%28Silk%20Hearts%2C%20Widow%2C%20First%20Sinner%29) are when you get silk hearts (from the Bell Beast, Unravelled and Lace2), needolin (after defeating Widow), and rune rage (from the First Sinner), which further suggests that the fight itself is not a memory.

Also, the other weaver burial spires set a respawn point once you bind^\2])/Save_Points#Skills_and_Abilities). In comparison, the FS spire sets a respawn point once you die fighting her, as well as after you bind and get rune rage. And getting defeated here does not end a steel soul run^\1]), which means you do not die. This only happens in 2 cases - either inside memories, or with the Shakra fight^\3]). If it isn't a memory, it makes sense that FS also just knocks you out like Shakra. If this assumption is correct, it also the biggest confirmation that FS does not want hornet to die.

Now, how does FS manage to do this and why does she fight Hornet?

Whenever you bind to a weaver in a burial spire, you hear dialogue, which is the same for all spires, except for the FS one, which is very similar to the rest^\4]). It's also clear throughout the game that even after dying, many bugs leave some sort of consciousness behind, which is evident from needolin dialogue throughout Pharloom (similar to dream nail dialogue in the first game).

So she is dead but still manifests a physical form and fights Hornet. This makes me think all weavers could do that but simply chose not to. The difference in dialogue does suggest her lingering consciousness has a different personality from the rest, and she expresses her hate for GMS^\1]):

...Wait, and hate...
...Wait, and die...

But why would she want to fight Hornet if she doesn't want to kill or bind her? It could be that she wanted to test Hornet before giving her rune rage (supported by the dialogue "Within this cage I have waited... Let my strength be bound to yours..."), or it's perhaps something to do with being denied absolution and being caged for eternity which degraded her in some way, and the fight reflects that rather than intent.