Is it safe for a person who underwent a back surgery to do bungee jumping?

Doubtful! Bungee jumping isn't what i'd consider a safe sport under any circumstances! especially if you have any type of hardware in your back (screws, stabilizing rods), those could easily be dislodged or become unstable. I would strongly recommend that you find another activity that does not involve strong gravitational force. Bungee jumping could put excessive pressure on your surgical repair.
Yes once healed. Once you are healed and cleared to return to activities and sports by your surgeon there should be no issue as long as the strap is intact but then it would not matter anyway.