Yes, Big the Cat was in fact playable in the Multiplayer mode of the original Dreamcast version of SA2 instead of the Dark Chao! He even uses Eggman’s mech walker.
I’m guessing Sega ran out of ideas for SA characters to add to the 2P mode. So they decided to add Big (he’s also on the dark side... wtf?). Then while making the ports, they realized that the Dark Chao would make an obvious fit.
To unlock him you must clear all of Eggman’s stages / missions with "A" ranks.
Sonic Adventure 2, HD, Remake, Battle, Dreamcast, DC, GC, Gamecube, Xbox 360, PS3, PlayStation 3, Big, the, Cat, Egg, Eggman, Mech, Walker, Fat, Froggy, A Rank, Robotnik, Doctor, Dr., Fish, Fishing, Tails, Miles, Prower, Tornado, Chao, Dark,Sonic,Hedgehog, Shadow, Amy, Rose, Knuckles, Echidna, Rouge, Bat, Shooting, 2P, Versus, 2-Player, Multiplayer, PAL, NTSC, American, European, Europe, Japan, Japanese, Easter Egg, Secret, Sega, Yuji Naka, Ryan Drummond, Rare, 2000, 2001,