Historically games were traditionally packed-in with games from the launch of the system until the system became obsolete. Often times a single title would serve as the pack-in for half the lifespan of a console with another title serving as a different pack in afterwords. (for instance the Sega Genesis had three pack in titles throughout it's lifespan: Altered Beast, Sonic the Hedgehog and Sonic 2) However starting with the 32 bit era this thinking began to change. Sony's PlayStation was the first console to not have pack-in game but rather come with a Demo Disc that had multiple game demos on it. The Nintendo 64 didn' t launch with any software packed in. Initial shipments of the PlayStation 2 also forwent a demo disk.
What seems to be the norm these days is for a console to initially come out as a barebones package, and then eventually get pack-in bundles down the road. A notable exception to this would be the Wii, which has always come with a copy of Wii Sports.
The Xbox 360 has a history of releasing game-themed consoles that have colors and decals that match the game it's packed-in with. Games such as Gears of War 3, Halo 4, Modern Warfare 3, Kinect Star Wars, & Resident Evil 5 all have console bundles that have custom coloring & controllers that match the game. Although the PlayStation 3 has plenty of game bundles, it does not typically do custom consoles, with the exception being the "gun metal black" colored console for the Metal Gear Solid 4 bundle.