Space Channel 5/WMG

The Extra Reports in both games are Evila's Dying Dream
In the first game, the Extra Reports are unlocked after you defeat  The Extra Reports are harder because Evila is the "Ultimate Reporter", and she's basically dreaming that she is Ulala. Now, what about the second game, where not only is  Simple. This explains the name changes like "Padding" and "Texas"; she just didn't know their names at the time.

Ulala and President Peace are both descendants of Giselle
It's clear during "That's How You Know" that Giselle possesses the ability to use song and dance to mesmerize crowds, and the crowds instantly know the appropriate moves to any dances Giselle may perform. Crowds behave identically whenever they're in Ulala's presence. And whenever President Peace sings. Obviously, Giselle's singing and dancing powers split eventually, and Ulala ended up with dance and Peace with sing. Plus, Ulala gets her pink hair from Giselle, even if it's been diluted from Giselle's red somewhat.

Fuse is the ship.
After all, we never see his face, and (unless I'm mistaken) he's always inside that blue-and-white Channel 5 ship...
 * And his voice change in the second game was the result of a tuneup!

Purge Jr is Purge who went through a time paradox.
He went to the future, and became his own son, a mistake he couldn't reverse. That's why Purge and Purge Jr look exactly the same.

Ulala and Purge are brother and sister.
Only two characters in the series can control Dance Energy, Purge and Ulala (I'm not counting Peace, he controls the Power of Song). Perhaps they're related, and therefore know how to use the Dance Energy. It would also explain why Ulala's so shocked when she first sees him in report 4 compared to Fuse's reaction.
 * Alternatively, Ulala and Purge were test subjects, genetically engineered to be the best dancers in the universe. But the overload of dance energy upon success caused the disaster that destroyed the space station. While Ulala was rescued by Jaguar and went on to be a reporter, Purge ended up going into robotics...

Purge Jr is Purge himself who suffered from head trauma.
Being sent into space might've had him hit his head. The trauma caused him to think that he's Purge Jr, not the original.

The game takes place long after the events of Elite Beat Agents.
After the Rhombulan invasion, the people are beginning to see the benefits of music-related technology to help against further attacks. Eventually, things powered by music became more and more common since the public appearance of the Agents, causing them to go out of business. Unfortunately, they didn't know that there would be ways to use The Power of Rock against the people. And the colonization of Venus? All thanks to the Zaiko Mega Corp. For added fun, President Peace is actually a descendant of JIN2. (You might need some Brain Bleach now.

Micheal transferred to Space Channel 5 from the Men in Black.
And President Peace is descended from the same alien race as Elvis.

Purge's armband is a power limiter.
He never takes it off. If he did take it off, his own power would be too much for him and he'd die. That's why he has to absorb people's energy to increase it, instead of taking off his armband and using his full power.

President Peace is a descendant of Dr. Ivo "Eggman" Robotnik
Irony points for being used to power a robot, then.