Is there a concept of a "sensation of self/me/mine"?

Is there a term or concept for 'feeling' like there's a "me", or for 'feeling' like something "belongs to myself"? Recently I was able to distinguish between two types of feelings, one where the feeling is just there and everything is fine, and another where a "me" is involved that "has" the feeling. The latter comes with a sensation, so I wonder if Buddhism has a notion of this sort of feeling that there's an identification or self that is involved.