Every Saints Row game is "of its times" in a sense.
Saints Row 1: Was developed as an attempt to make Grand Theft Auto San Andreas on Xbox 360. The Third Street Saints even were originally colored green. The game was largely an attempt to capitalize on 2000s era rap culture.
Saints Row 2: Was a direct continuation of 1. Was an even bigger attempt to capitalize on 2000s era rap culture esp in its marketing which prominently featured stuff like grills. The developers tried a bunch of then trendy ideas like ingame advertising that companies could rent out
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Saints Row 3: Was the first game definitively lampooning GTA4. They deliberately made it to parody the game's serious atmosphere. Like adding WWE-esque moves for the player, the Dildo Bat, and making a gang based on Luchadores. It was also heavily slashed back by THQ's bankruptcy. It was also full of minor dlc that was common at the time
SR4: Was a DLC that was made into a full game. Also was made to capitalize on the popularity of super heroes at the time
SR5: Made to appeal to Tiktok zoomers and fortnite players. Art style is made simpler and they add buzz words to all the lines like "late stage capitalism".
It's actually surprising how every game encapsulates a certain era of media so perfectly. Grand Theft Auto by comparison always felt totally separate.