Perhaps consciousness is shared, only memory isn't, and the distinction is mere an illusion due to distinct memory. Perhaps that's how aiming to be "self-less" is a thing in Buddhism, Sufism, Zen, etc.
Makes sense if you think about memory in the same way you think about time, as creations of the mind.
I read somewhere once about a story written in a book. The beginning and the end is there, everything superimposed together. For the characters in the book, there is no time.
Then the reader's mind/consciousness comes along and goes through the book, page-by-page. By this very act he creates memory and time. Maybe our minds are doing the same thing on an infinite, timeless substrate that Buddhism, Sufism, Zen etc call awareness.
I'd say perception is more relevant to the experience of unique identity than memory; I only see out of my eyes, which makes me feel separate from those around me, even if we'd have lived through all the same experiences.
I'm not sure perception and memory are all that different. What are memories except echoes of perception? I actually wonder if memory is not our perceptual signals caught within neural loops that are able to feedback and trigger pattern detection circuits. When this happens for multiple patterns in synchrony, we form associations. Is it right? Who knows.. no idea how even neuroscience can start to answer that..