See, I began playing Saints Row back when it was brand new. The first game was a plain GTA clone that got attention because of its colorful graphics (especially since this was when the dull browns and grays of faux-realism began to hit it big) and actually being a good game by its own right. Biggest problem was probably the very bare bones story, with the protagonist speaking exactly once in every gang finale mission. There were still typical colorful characters and the "dirty parody" humor style, like a Wendy's restaurant being named "Freckle Bitch's." But despite the polished and fun gameplay, it was basically the skeleton of a larger game.
Saints Row 2 added to that and came out when GTA IV had taken the gaming world by storm with a much more realistic and serious take on the series, greatly subduing itself from the insanity of San Andreas. Saints Row 2 added a talkative and more customizable protagonist and tried to add on to the storyline (as well as connecting it even more to the Red Faction universe by showing the rise of Ultor, turning a throwaway joke into something much more serious), but it mainly got by as an alternative to the brooding philosophizing and subdued humor of GTA IV. It was bright and colorful, let you use a minigun to clean up the streets for a COPS-esque TV show, included a long side mission involving spraying feces from a truck at various houses and vehicles, and had almost cartoonish physics.
Saints Row 3 took everything that the developers figured people wanted and turned it all up to 11. The cartoonish element and plain denial of the laws of physics and sometimes reality itself were brought to greater extremes, almost like it was a parody of itself. Saints Row 2 had Yakuza, punk-style maniacs, and a Caribbean/Voodoo drug syndicate. Saints Row 3 had luchadores and cyberpunk hackers. This was the point where I got really disinterested, because it seemed to care more about "not giving a fuck" than anything really cohesive.
Saints Row 4? Well, we're seeing superpowers from the first seconds of the trailer. I'm not even sure how close this game is going to be to its original incarnation, and I'm not sure if the developers have really had any kind of plan for the series starting with the third game beyond "Turn up the dial."