Pretty much every console we've ever owned has been 'shared' for the family. Even the 360 and PS2 I bought for myself - the only two home consoles I've ever purchased - have been family consoles. I had to share with not only my three older sisters, but my parents as well. I still remember coming home from school and seeing my parents sitting on the living room floor, playing Lolo on the NES. {XD}
That said, I grew up with an Atari 2600, Intellivision, and NES. They added an SNES and N64 to the mix when I was still pretty young, so you could say I grew up with those, too. But the first three listed were my "first" consoles.
The only consoles that have ever been exclusively mine were my handhelds. Specifically, I received a yellow Gameboy Color for Christmas - the same as many others here, it would seem.
My sisters had no interest in them after passing down their original Gameboys to me, and my parents had complaints of being too old to use the tiny screen.