Once the organic material in a cyborg dies, it either needs to be replaced, or the cyborg dies. You cannot keep the dead tissues working; if you could, they wouldn't be dead, after all.
How does the cyborg maintain his human identity? Well, as long as his brain is still functioning, you might as well ask how an amputee maintains his humanity. This would be a highly philosophical question, but scientifically speaking, your "being" is in your brain.
So what if the cyborg's brain dies? This leads us back to the above statement: He either replaces his brain, or he dies. There are of course different ways he could replace his brain:
Essentially, as long as you keep cloning your original tissues, nothing else matters. Only, a human's "being" isn't made of only his brain, but is also formed by knowledge and experience. Or in other words: External influence. To maintain his human identity, the cyborg would need to somehow copy all information from his old brain to the new one; a backup of sorts.
A new brain
Here, your cyborg just takes one of several brains available on the market. Morals and ethics aside, as long as the new brain is compatible with the cyborg's inorganic components, there's nothing wrong with using a different brain.
Just like with cloning, the cyborg would need to backup his old brain first. He'd also need to wipe his new brain clean, lest he wants to enjoy his newfound schizophrenia. And finally, he'd need to ensure his new brain works the same way as his old one, or else he might behave differently to certain stimuli, which would arguably impact his identity.
When old tissues die, what better excuse to replace organic components with inorganic ones? Using a powerful microchip instead of some inefficient and highly sensitive brain certainly has its advantages; immunity to concussions being one of them.
In order to maintain his human identity, the cyborg would not only need to restore a backup of his old brain, but he'd also need to ensure the chip emulates his old brain as closely as possible. Naturally, he'll eventually adapt to the fact that he no longer has to hold back when head-banging, which would impact his identity, but that's inevitable.
Once the cyborg's using a chip instead of a brain, you might ask yourself if he's still human in the first place. That is also a philosophical question. In fact, it's not just a single such question. There's the Ship of Theseus which is about the question: If you keep replacing small parts of a ship until none of its original parts remain, is it still the same ship? And there's also the question: What does it mean to be human? Can a machine be human as long as it behaves like a human?
Regardless. As long as the new brain holds all of the information of the old one and is capable of processing those information the same way the old brain did, your cyborg should retain his identity at the very least. Whether he's still himself or even human in the first place, that's something you'd better ask the philosophers.