The 'World Tag Team Championship was established on November 14, 2001, the night after the WCW Invasion ended. WCW Tag Team Champions, the Dudley Boyz had unified the WCW and WWF Tag Team Championships by defeating the WWF Tag Team Champions, the Hardy Boyz, the night before. The Dudleyz were considered to be the last WCW Tag Team Champions, and the new WWF Tag Team Champions.
When the Tag Team Championship became the exclusive property of Monday Night Raw, then-Smackdown! GM, Stephanie McMahon introduced the WWE Tag Team Championship. At this time, the WWF/WCW Tag Team Championship became known as the unbranded "World Tag Team Championship." This was partially done so that its name could mirror "World Heavyweight Championship" on Raw, while Smackdown! had the WWE Championship and similarly-named WWE Tag Team Championship.
When various champions started getting drafted between brands, the two sets of Tag Team belts always managed to remain on their "original" shows.
Currently, the World Tag Team Championship is a part of the WWE Unified Tag Team Championship.