The Dark Brotherhood chooses black, skin tight and flexible gear because it makes it easier to move and easier to hide in the shadows.
It probably acts like a poor water suit as well, which allows them to last a little bit longer in the freezing waters of Skyrim without freezing to death.
They don't care about protection, because they're never supposed to get hit anyway.
They don't care about being obvious, because when they're wearing an assassin suit in the first place, they're obviously not going to try to mingle with the crowd. They are supposed to sneak in the night and not be seen in the first place.
I'm sure each and every member of the Dark Brotherhood probably have a set of common spare clothes to put on if they want to mingle with the crowd.
But... I agree, the armor, despite it's practical uses, looks silly. I never understood the red colours