The game is basically a prequel to Red Faction. Here's a couple of the Shout Outs to the game.
The Matsuko army.
A radio ad is for a show based on the game.
A radio call in a secret lab for Dr. Parker.
The Saints make a movie which is basically Red Faction.
The Saints are something of a homage to the New Orleans Saints.
In the intro scene that serves as an advertisement for Saints Flow, Pierce clearly channels Goku when he shoots energy balls out of his hands and even does a kamehameha wave!
Except for the part where it more closely resembles Hadouken. And according to the Saints Row Wiki, the name of the weapon is "Quarter Circle Punch" in a DLC that gives you that power. At the same time, the taunt for it is vanilla and is called "Quarter Circle Forward + A".
In Saints Row IV, if you give the "Hadoken" taunt to Female Boss voice 1, she says "Spinning Bird Punch". A reference to Chun-Li's spinning bird kick and the fact that the voices for Female Boss voice 1 and Chun-Li are one and the same.
In Saints Row 3's Whored Mode, one of the waves on the Zombie Island map gives you a chainsaw against a bunch of zombies. The wave is called Hank East and the subtitle is "He's covered wars ya know".