A grey card is made to reflect all colours equally. White paper often is not: many papers contain brighteners that glow bluish, and a white surface easily turns into pure white in the photo, which leaves nothing to measure. A grey card keeps detail at normal exposure and stays neutral, so the reading taken from it is reliable.
The card is placed where the product will stand, angled like the product's main face, and photographed once in the same light. The camera's custom white balance can be set from that frame, or the white balance tool of an editor is clicked on the card afterwards, and the setting is copied to every photo from the same light. When the light changes, a new card shot is taken.
The card also helps with exposure. A camera meter aims for middle grey, so a white product on white is shot too dark and black on black too bright. Metering from the card gives the correct exposure for both. For matching colours across several cameras or studios, a chart with colour patches goes further than a single grey card, which only sets the neutral point.