1. A method for efficient FlashCopy backup target volume allocation with reuse and a shared resource pool by a processor device in a computing environment, the method comprising: dynamically allocating a new target volume for a new FlashCopy backup by performing at least one attempt to reuse an existing FlashCopy target volume containing older FlashCopy backups that have been previously ingested into a repository and are not currently being used as a source for at least one of a mount task, a clone task, and a restore task; wherein, if the at least one attempt to reuse the existing FlashCopy target volume fails: dynamically allocating the new target volume from a global pool of FlashCopy backup target volumes.