Looking at some of the ships in this presentation, you will see that some are actually repurposed ships. This applied to any available ship that could be readily adapted for use in the fleet and reduced the time it took to build one. In the case of a tanker and a Fleet Oiler, the differences are few. The ships are basically the same, but the tasks they do are different.
The Fleet Oiler's job is to keep ships at sea and extending their deployments. This was actually only part of a procedure called UNREP or underway replenishment. With the demands of a ship needing to be at sea, it was impractical for the ship to return to port for fuel. This was one of the key factors in winning a war. What actually became commonplace was taking a tanker and making it able to refuel ships at sea. The actual practice was put into place before WWII. Below is the USS Ramapo, AO-12 which was an oiler that served in WWII. The more modern USS Cimarron is pictured above.
How did the process work? A line was shot across to the ship being refueled. The line would be secured and fueling hose would come across on the line. It was very important that the ships maintained speed and that a wave didn't make the line go too tight, and snapping it. What made this so dangerous was the risk of severe injury if a sailor on deck got hit by the line as it parted. In some cases, it killed sailors ad they were cut in half or had legs amputated.