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izuku places his entire sense of identity and self-worth on being a hero, and having the quirk that allows him to be one. those two things, heroism and having this quirk, are inextricably linked to him - when he was quirkless, he couldn’t be a hero, and he got the quirk so he could be.
izuku also clearly identifies heroism as saving people. when he fails to save people - ie fails at being a hero - he feels worthless. if he doesn’t save, he’s not a hero, and thus is nothing.
now apply this reasoning to when he offers mirio his quirk after the overhaul arc. again, without a quirk, he can’t be a hero. why is he wanting to give away the very thing that he thinks makes him valuable to other people?
because he failed to save someone, so he’s already not a hero. he thinks he doesn’t deserve his quirk, the thing that gives him that worth and identity, because he doesn’t deserve to have any worth or identity. he’s already squandered it. he deserves to be nothing again.