It is a "trick riddle" with the sole purpose to get the answerer to say something that the asker can "hear" as an invitation to pinch, poke or beat the unsuspecting victim. Especially children find this funny.
Haumich, while seemingly used as a name, is a contraction of "hau' mich" (= beat me). Another variant of the joke starts with
Zwickmi (= pinch me) and Zwackmi (likewise) gehen in den Keller...
Pflaumich is a nonsense word, probably choosen just for the rhyme with Haumich, so "mein Pflaumich" does not make any sense in standard German. I can not exclude a regional term though.