Wine.
I’m not a sound, though I sound like one. “Wine” and “whine”.
I come from somewhere that sounds the same. Come from the “vine”.
With the staff of life I am much desired. “A jug of wine, a loaf of bread - and thou.”
Poetic? See the Rubáiyát of Omar Khayyám.
Two rhymes in the clues: “same” and “name”.
Two rhymes in the answers: “wine” and “vine”.
Homophone to the onomatopoeic answer to the first clue. “Wine” and “whine”.
Drawn from the earth by root, sun and rain, wrung from the bounty and hidden again, “cooled a long age in the deep-delved earth”, fleeting of pleasure but easing of pain, share to bring close and friendship to gain.
OK, sorry, I gave you a whole new riddle with the same answer. The vine draws from the earth, the grapes (bounty) are “wrung” and the juice “hidden” in casks, “cooled a long age in the deep-delved earth” is a quote from Keats referring to wine, wine gives “fleeting pleasure”, but has been associated with transcending suffering in literature and history, and sharing wine brings friends together.