Grey and white are the same color to the camera. You can consider grey as dark white or white as light grey. So it is very much the same and you can take a white-balance measurement from any shade of white as long as it reflects light uniformly. As you read from the answers you already got, white paper is not necesarily white.
Depending on your camera, the custom white-balance function either uses a set or an automatic exposure. In the former case, using a bright object may cause it to be over-exposed while will either give you an error message or an incorrect white-balance reading. In the latter case, the camera would be able to meter on any shade of white and, if you will the frame, even a white-paper will appear roughly 18% grey.
If you buy an exposure card in a photo store, they will sell you something 18% grey. If you buy a white-balance card, it will be closer to 80% grey, so almost white. You can buy combined products, like the one I use which has one dark side and one bright side. The difference exists because a digital camera is able to make a more accurate measurement with brighter values. Dark surfaces are more susceptible to image noise which often introduces a color-shift because you may have more noise on one channel.