He doesn’t mean “Sorry” in terms of wanting to die. He’s not truthfully sorry about that. He can’t be, because he really does value himself that little.
But he means the apology when it comes to how much he’s disappointed Kurogane.
This is like back in Outo, when Kurogane expressed his anger over THIS EXACT SAME ISSUE ACTUALLY. Kurogane criticised how little Fai valued his life, and at the time Fai misinterpretted it as Kurogane saying he hated him.
And here again, Fai offers to die and KUROGANE IS, AGAIN, IMMEDIATELY MAD.
But that’s all Fai latches onto.
He sees the anger and knows he caused it. He sees the vehement disappointment and knew all along that he was going to disappoint this man.
I don’t think he properly sees the reason why Kurogane is so angry. I don’t think he sees how furiously Kurogane cares for him and, obviously, still doesn’t want him to die. Kurogane wants him to WANT to live. Kurogane wants him to TRY.
But then again, maybe Fai does see it. On some level, at least, he must know.
And maybe he’s apologising for the fact that Kurogane cares so much for him when Fai doesn’t think he deserves it.