Giving up a Zechut to Uplift a Neshama

When one performs a Mitzvah liluy nishmat for someone else, does the person give away their zechut, or is there now a zechut for both people?

Not at all. The only time there is a discussion like that is either when a person makes a pact to support someone in exchange for part of their mitzvot and torah, or he openly says he gives up his zechut. Even then most poskim hold he doesn’t lose his zchut.