I think I understand what you mean.
You mean that you see "a part" of yourself in everyone you meet.
That is a nice way to say that all humans have similar lives, needs, desires, and fears.
- we all need to eat
- we all have happy times
- we all have sad times
- we all have good health sometimes and sometimes poor health
- we all want to have peaceful lives
- we all love and we all cry
So, in many ways, all humans are similar.