Here the speaker treats their own mind like a troublemaking neighbor, one they can't escape because they're joined to it like a siamese twin. The humor comes from that image: the speaker cringes at the thoughts that pop up and jokes that if they weren't so closely attached, they'd suspect the mind of drinking.
In the second stanza the embarrassment turns into real worry. The speaker fears that the mind's "next new meme" will offend friends so badly that they'll cut the speaker off too. Being tied to something you can't control means you share the blame for what it says.
The last stanza is a kind of comic declaration of independence. The speaker decides to ignore the mind and refuse to "play its game," then protests, "I'm not the one to blame!" The mood is light and playful, with bouncy rhymes and a wink, though underneath it lies a familiar anxiety about unwanted thoughts and how others might judge us for them.