From a production point of view, the DS9 uniforms were created to give the show a distinctive look different from TNG. I think the original intention was that the DS9 uniforms were meant as a utility uniform, something officers would wear in a more 'gritty' environment.
The reason VOY inherited the DS9 style uniform instead of the TNG was a cost saving measure. The TNG uniforms were getting worn out, so rather then creating new ones, they could 'split' the uniform costs between both shows.
When DS9 switched to the First Contact style uniforms, VOY inherited all the DS9 uniforms. I also believe that the costs to make the new style uniforms for the DS9 cast, was absorbed into the FC budget rather than DS9's budget.
One piece of Trivia: Star Trek Generations was supposed to debut a new Starfleet Uniform but apparently they looked bad on screen so they switched back to the TV uniform. However, Jonathan Frakes DS9 uniform was actually Avery Brook's and Levar Burton's was Colm Meaney's. Only Patrick Stewart and Brent Spiner got custom designed DS9 uniforms.