Having a memory is not a necessary nor a sufficient condition for possessing a self-identity.
We are back to square one. The poor souls in Oliver Sacks' tome, "The Man Who Mistook his Wife for a Hat" are unable to create and retain memories. They occupy an eternal present, with no past. They are thus unable to access (or invoke) their self-identity by remembering it. Their self-identity is unavailable to them (though it is available to those who observe them over many years) - but it exists for sure. Therapy often succeeds in restoring pre-amnesiac memories and self-identity.
V. The Incorrigible Self
Self-identity is not only always-on and all-pervasive - but also incorrigible. In other words, no one - neither an observer, nor the person himself - can "disprove" the existence of his self-identity. No one can prove that a report about the existence of his (or another's) self-identity is mistaken.
Is it equally safe to say that no one - neither an observer, nor the person himself - can prove (or disprove) the non-existence of his self-identity? Would it be correct to say that no one can prove that a report about the non-existence of his (or another's) self-identity is true or false?
Dan's criminal responsibility crucially depends on the answers to these questions. Dan cannot be held responsible for Jack's murder if he can prove that he is ignorant of the facts of his action (i.e., if he can prove the non-existence of his self-identity). If he has no access to his (former) self-identity - he can hardly be expected to be aware and cognizant of these facts.
What is in question is not Dan's mens rea, nor the application of the McNaghten tests (did Dan know the nature and quality of his act or could he tell right from wrong) to determine whether Dan was insane when he committed the crime. A much broader issue is at stake: is it the same person? Is the murderous Dan the same person as the current Dan? Even though Dan seems to own the same body and brain and is manifestly sane - he patently has no access to his (former) self-identity. He has changed so drastically that it is arguable whether he is still the same person - he has been "replaced".
Finally, we can try to unite all the strands of our discourse into this double definition:
It would seem that we accept that someone has a self-identity if: (a) He has the same hardware as we do (notably, a brain) and, by implication, the same software as we do (an all-pervasive, omnipresent self-identity) and (b) He communicates his humanly recognizable and comprehensible inner world to us and manipulates his environment. We accept that he has a specific (i.e., the same continuous) self-identity if (c) He shows consistent intentional (i.e., willed) patterns ("memory") in doing (b) for a long period of time.
It seems that we accept that we have a specific self-identity (i.e., we are self-conscious of a specific identity) if (a) We discern (usually through memory and introspection) long term consistent intentional (i.e., willed) patterns ("memory") in our manipulation ("relating to") of our environment and (b) Others accept that we have a specific self-identity.