Officially, the short answer on if the game has a damage meter of some sort is no. However, that is just inside of normal dungeons and raids. There is a DPS meter that players can use inside of the training room. You can use this one to gauge your skills. But once players leave the training room, that DPS meter will no longer be accessible.
There are alternative ways on getting a DPS meter working inside of normal dungeons and raids. Though, all of these different ways would include installing unofficial tools so we would highly recommend not using them. Using unofficial software can lead to save corruption and other nasty types of file corruption.
The topic of including a damage meter has been passed around since the game officially released. Many other MMORPG titles have included them such as World of Warcraft. However, many believe the visibility of seeing other players’ damage per second leads to toxic online environments.
