Chopped liver insignificant? Oh-no-no! (Especially during the digestive process
No - the reason chopped liver is used in the analogy (correct usage, "So? Vat am I? Chopped livah?," accompanied by half shrug) is that it is omnipresent*, therefore easily taken for granted. A subtle, but important difference. In correct usage the saying implies the sayer's sense of self-significance and conveys an appropriate load of guilt for one so thoughtless as to forget. The sayee is to infer that he/she is real schmuck for taking one so significant for granted.
*Or was, in the old days, before The Most Expensively-Catered Bar Mitzvah Spread Wins (shhh - don't mention we got wholesale!) tradition got started.