If so, please note the difference between Shaded and Rendered viewport modes.
Custom Preview will preview the colors in Shaded mode but not bake the object colors.
It will also preview the colors in Rendered mode and when you bake it, it will add new materials for the objects representing the colors you specified.
These colors will show up in Rendered mode because they are materials but not in shaded mode by default because shaded mode is set to show “Object Colors”
Graph Space: