> All points in my body are equally part of 'me'.
That's a reasonable belief, but some people do not feel that way.
If I had my foot amputated am i still me? If i had my mind transplanted into a different body am I still me? etc. I doubt there are any good answers that are more logical than "I feel this to be true".
> I don't understand your reference to the western world
Some religions believe that people have a soul that is separate from the body, which contains a person's essence (And say, goes up to heaven when they die or otherwise continues on after death). Even among people who don't buy that, it sets the stage (even perhaps just subconciously) for other dualist beliefs where the body is separate from your "essence" (however you want to define that)