Yea, I checked it up on Bulbapedia and the game data has this to say: "By moving rapidly, the user makes illusory copies of itself to raise its evasiveness."
Basically, what your Pokemon is doing is moving so fast, it creates afterimages of itself. Or it glows white and makes the copies if they aren't that fast to begin with. The times a Pokemon has ever made copies that didn't disintegrate after being hit or phasing through the opponent when "attacking" was Greta's Umbreon in the Pokemon Adventures manga, and the manga had other Pokemon using Double Team that weren't solid to begin with. The only manga that had this consistently was the Pokemon Pocket Monster's Manga, but it was only during two battles between Red's Clefairy and Tyrogue against a trainer's Dragonair, and an unknown trainer's Gothorita in a later chapter of the Pocket Monsters BW. Since then, it's been established that the copies from Double Team are illusions. The only Pokemon in the anime that ever used Double Team and had "physical" copies was a Deoxys in the Destiny Deoxys movie, but that was a Mythical, and no Pokemon is ever on par with a Mythical or Legendary in terms of power.
Basically, none of the copies should be physical or solid to begin with, and none of them should affect the original or should trip on each other because they aren't real to begin with.