The feeling off "self" is not an illusion, we all or most of us have it. But it is just a word and one whose meaning is not very well defined, it is like you say "whatever it is". Which kind of hints at that we don't really have a good definition for it.

I am "me" and the definition of "me" is what I am. Somewhat cyclical right?

If you lost all your thoughts and memories, what would be left? You can call that "self" but is it the same "self" as what you thought of yourself as before losing all thoughts and memories?

There are people suffering from schizophrenia who alternate between different "selves". At least to their experience those are totally different "selves" which need not be aware of the other selves at all. So would it mean there are multiple "selfs" within one brain?