Your "self identity", as in, the "image" or "character" that you imagine yourself as, the "person" that pops into your mind when you think "this is me", is merely a mental construct. It is an amalgamation of all the things you have done, or were done to you, or people you've been close to (family, friends), or media (books/games/movies) you've consumed and "related to" (internalized as a part of yourself).
You can formulate this "avatar" or "person" by describing all the properties it has. Lie, try to describe "who you are" right now. "I am a person who was born at such time in such place, to such parents, I like such and such things, I believe in such and such ideas. I have such and such persons who are involved in my life. My plans/hopes for the future are such, and they stem from experiences I had in X age, when I was first [whatever]..."
The amalgamation of all of these thing is what you tell yourself about yourself. Or rather, the amalgamation is a sort of "avatar" or voodoo puppet that you invest your "identity" into. This collection of memories, experiences, people, events, places, etc., is what constitutes this total definition of "me" (You).
This Frankenstein's monster of concepts and ideas and events is what you see as the thing that defines what is "you". Just like a definition of a word in a dictionary is a collection of other words that form a meaning that combines all of the words, the definition of what "you" are is a collection of things and memories and events and things you associate with "yourself", that comprise the ultimate Meaning of "You".
The trick is that this mental construct of events and ideas, this description of "You" is completely arbitrary. It's just a collection of memories that you simply gave a label to, with "Me" written over it. It could be anything, it's just an arbitrary collection of memories and things.
If you're the same age as me, maybe you remember "emos" or "punks" or "real pacani", or whatever other idea some people associated with themselves at the time. Same with movies and music they liked. "This pop song I like is so
me'". "This fashion style I like is so
me'". "This movie I like is so `me'". And so on.
And if you're anything like the average person, you eventually "grew out" of those things, and they no longer define who you are.
But think about it. You switched one set of objects that used to define "you", for another set of objects that now define "you". If you can just change the definition of "you" through your life, like as if you simply rewrote the definition of a word in a dictionary, is there really a "word" there?
Is there really a "you" behind your "mental possessions"?
Many people go through life with some thing that threads all of their live events together, creating an illusion of a "story of life" or a "narrative" or "meaning". For some it's parents, for others it's a video game they've been playing since childhood (I first played counter-strike 1.6 as a kid, I still play CS:GO as an adult = "I am a Counter Strike person"), for others it's a genre of music they're into, whatever whatever. And that's their "identity".
OFC it can be more complex than that, usually it's multiple threads of events and people and objects that ultimately end up defining what the person is "about". But I hope the examples are a good enough illustration of the basic concept.
So, if people define themselves by certain "things", and also sometimes abandon those "things", and adopt new "things" that are "them", the question is, is there an actual, authentic "real person" underneath? Is there a "unique human individual" not defined by the circumstances of their birth, the things they consume or consumed, the people they just happened to meet in life, etc. etc.?
I claim no. There is no central, unshakable, unwavering "SELF" behind every person. We are only that what we "adopt". Which means that there is no fundamental "self" at all. On the deepest, most fundamental level, we're just observers looking at what is happening to our bodies and "personas", as the bodies go through life, autonomously, following their instincts. Essentially, we don't have "souls". Merely parameters that generally define our behavior, and the "history" of things that we did, or others did to us, as a result of those parameters. That's it.
We're along for the ride, but we're not at the steering wheel, so to say. The smallest, irreducible "thing" at the center of every person that I can see so far, is the "observer". The Mind's Eye. As in, the thing that notices all of the things that are happening to oneself, and the thing that notices that it is noticing.
If you wish to get more in tune with the "observer", or the "eye that notices", I recommend picking up meditation. Maybe of the mindfulness kind. Such as this: https://www.youtube.com/watch?v=CN-_zzHpcdM
Sam Harris is a retard, but he made a pretty good meditation video. You should check it out.
Anyway. All of the above was said from the perspective of the "self", but also applies to the perspectives of others who are looking at your "self". As in, if you have an internal idea of "who I am", then others also have an idea of "who you are". You might think of yourself as X person, but that one cashier who sees you sometimes every week buying groceries has her own internal "avatar" or "mannequin" or "scarecrow" or your "being". You might think of yourself as "such and such", but the cashier lady sees you as "that guy who buys this and looks like this and does this". And those two entities are completely separate and irreconcilable. Some stranger's idea of who you are is as valid as your own idea of who you are, because both are mental constructs.