The ship of Theseus doesn’t help your cause here. I’m not the same person I was 5 seconds ago. The self is obviously mutable, which kind of makes sense as long as there is a continuity of consciousness.

What puzzles me is the persistence of identity across interruptions of consciousness, even though your brain can be mutated while you are not there.

The only mostly permanent thing in the brain is neuronal DNA since most neurons don’t undergo mitosis.

IIRC consciousness is about a regime of brainwaves, while memories (and therefore "continuity" of consciousness) are about the actual neuronal connections, and so "survive" sleep ? (Though sleep is also known to be used to integrate new memories, so in that sense a (slightly) different person wakes up each time.)