Let me see if I got this right.
Jak & Daxter vs. Ratchet & Clank: Both argue who's better.
Nathan Drake vs. Sly Cooper: Nathan's ring is valuable and Sly tried to steal it.
Fat Princess vs. Evil Cole (WTH?): I'm guessing a fat joke is the reason.
Sir Daniel Fortesque vs. Radec: The Helghast Colonel makes fun of Sir Dan for being a terrible leader of his king's army by being the first one to die right before his game even started, pissing the skeleton off.
Nariko vs. Dante: The Angel/Demon hybrid annoys Nariko with one too many pick up lines. That or Nariko sees him as an abomination for being something angelic but with demonic powers and shouts something along the lines of "kill it with fire". First one would be kinda humorous but I think the latter would be hilarious because imagine if New Dante were to represent the positive side of the DMC fanbase while Nariko sides for the exact opposite that despises him. Given the large amount of love and hate for the westernized version of the demon hunter on YouTube, I don't think Ninja Theory would be able to resist making such an easter egg.
Spike vs. Parappa: Spike believes Parappa is working with the monkeys.
Cole MacGrath vs. Raiden: Maybe Cole accidentally siphons electricity out of Raiden's cyborg body, making him kick start a backup generator inside him or something.
Sweet Tooth vs. Kratos: Because they're both guys who offed their families and spend most of their time going on murderous rampages, the clown will probably try to make the spartan revel in his actions and embrace the madness so he can have a BFF around but all Sweet Tooth does is make Kratos angry.