This will come as a huge disadvantage for competitive battling if it did exist. Pokemon with the same typing for two types would always have a 4x weakness, which certain attackers can capitalize on. It's too silly to even think about it when you consider that Grass/Grass and Rock/Rock have 5 double weaknesses, and that any type weak to Stealth Rock would always lose 50% of HP upon entry. Moreover, it has the same STAB coverage as those of single-types. This idea doesn't even change the look of Pokemon. It it just never going to happen and these reasons above are why I wouldn't consider one for my team.
But think of it this way - surely it would also mean that you also get a double STAB bonus, SpecsOgre (..and SpecsToed I guess) just got that much more powerful, have fun tanking THAT Water Spout!
Double STAB 252+ SpA Choice Specs Kyogre Water Spout vs. 0 HP / 0 SpD
Palkia in rain: 215-253 (66.97 - 78.81%) -- guaranteed OHKO after Stealth Rock and Spikes, 37.5% chance to OHKO after Spikes
Double STAB 252+ SpA Choice Specs Kyogre Water Spout vs. 4 HP / 252+ SpD
Eviolite Chansey in rain: 469-553 (73.05 - 86.13%) -- guaranteed OHKO after Stealth Rock and Spikes, 81.25% chance to OHKO after Spikes
Double STAB 252+ SpA Choice Specs Kyogre Water Spout vs. 252 HP / 208+ SpD
Ferrothorn in rain: 338-398 (96.02 - 113.06%) -- guaranteed OHKO after Stealth Rock
Double STAB 252+ SpA Choice Specs Kyogre Water Spout vs. 252 HP / 144 SpD
Arceus-Grass in rain: 381-448 (85.81 - 100.9%) -- 87.5% chance to OHKO after Stealth Rock, guaranteed OHKO after Stealth Rock and Spikes
Double STAB 252+ SpA Choice Specs Kyogre Water Spout vs. 248 HP / 252+ SpD
Giratina in rain: 319-376 (63.41 - 74.75%) -- 75% chance to OHKO after Stealth Rock and Spikes
Just run spikes support and Tailwind and you'll be set for the sweep of your lifetime! I'm down.