Depending on interpretation, you could be a...
digital screen, such as computer's monitor display.
I am many colors, but I'm not a rainbow.
A screen produces graphics by using many many tiny pixels, which can change color. They form patterns that can represent text, images, and other visuals.
My skin is the cousin of chess, yet my board is extended.
The pixels are organized into a grid, with rows and columns like a chessboard. The word "extended" here could mean "dynamic", in the sense that a pixel can change color.
I am with you here and sometimes at home.
If you can see this (here), then you must be looking at some sort of digital screen. And many homes contain devices with digital screens, such as personal computers.