The phrase for what's going on in that Rolling Stones lyric is "Negative concord" and yes, it's normal in many languages and that includes a lot of non-prestige English variants.
"Ain't nobody got time for that" is clear, likewise "I didn't go nowhere" and "He ain't take nothing from nobody" and when something is not clear ("That ain't nothing" could mean it is, or it is not, something) context usually suffices.
"Ain't nobody got time for that" is clear, likewise "I didn't go nowhere" and "He ain't take nothing from nobody" and when something is not clear ("That ain't nothing" could mean it is, or it is not, something) context usually suffices.