Using a double negative instead of the negation of any is used often in everyday English, and by extension, English-speaking musical artists:
- Nina Simone's "Ain't Got No, I Got Life"
- Woody Guthrie's "I Ain't Got No Home"
- Bill Withers's "Ain't No Sunshine"
- The Frankie Miller Band's "Ain't Got No Money"
Outisde the musical community, the late Rodney Dangerfield was famous for his bit, "I don't get no respect," where he used self-depcrecating humor and an everyday demeanor to great effect.
So it's not unusual at all: the double negative is idiomatic of a native or fluent English speaker. It's not something the Chicago Manual of Style would recommend, but most English speakers understand exactly what is meant, and that's the most important thing in a natural language.
As to why it's particularly "I can't get no satisfaction," Keith Richards came up with it and its accompanying riff after waking up in the middle of the night, and falling back to sleep immediately afterwards. One can presume it's not the Queen's English because he wasn't wide awake.
However, you should notice "I can't get no" and "Satisfaction" are both four syllables long, which means the riff he came up with can be played on both parts. That wouldn't work if it was "I can't get any satisfaction" or "I'm unable to receive any satisfaction."