He dresses like a cop because his constituency fetishizes cops and the military. People in uniform look official and project some level of authority. Therefore, morons subconsciously lend him more respect and good will than they would if he was wearing a tan suit or something.
His constituency fetishize cops and military because: 1. they are told to in a crap load of messaging everywhere from the NFL to a crap load of stupid truck stickers you see on half the vehicles in our world, 2. they are careers that are easy to get into and that provide what they see as a path to respect, power, or authority without actually having to accomplish something difficult that would allow you to earn those things, and 3. they get to play with all the "eff yeah" equipment they like to mentally jerk off to(larping as if you are GI Joe).
Greg draws on that to make his voters want to trust him without consciously making that choice.