In Polish slang "słoń" do not have "secret" meaning. "Jego ptak jest duży" or "Ma dużego/wielkiego ptaka" suggest size of "penis".
Ok, so jego is either genetive (on, ono) or accusative (on). Which case is it here and why ?
„Jego” is normal possessive pronoun and it does not decline. To be precise pronouns: jego, jej and ich do not decline at all, but others like mój, twój, swój do. So if the pronoun was different we would use nominative here.