Could "go" here also refer to things that would in English be translated as "it", such as when it refers to a dog (since it's masculine in Polish)? Thank you!
none in meaning. "Jego" is used in accented position in the sentence (always at the beginning, often at the end if a sentence is longer) , or to stress the word, "go" is used only in not accented position in a sentence.
Accented position is also after preposition, but genitive=accusative form of "on" has "niego" for that.
As an English speaker (UK), I (and steven.van4?) imagine a crime story where an elusive suspect is finally arrested, and at the press conference the police spokesperson says "Ladies and gentlemen, we have him!" (or: "We've got him!").
A native Polish speaker might know other connotations of "Mamy go!"