Cleaning is a compromise between removing what you do not want to be there and keeping what should be there, in colored clothes you what to keep the color pigment but that is not a requirement for white fabric. In white fabric you need to remove more of what is there to keep the white. A stain that is clearly visible on white fabric can be invisible on colored fabric. So detergent for colored clothes needs to be gentler for them to keep the color.
An extreme example is a lot of bleach would run most colored fabric but make the white fabric look better.
Detergent for white clothes contains a bleaching agent that helps keep the fabric white. if you use them on colored fabric it can cause them to fade in color.
There is often optical brightening agents that make fabric that gets a bit yellow to look white, it is typically blue pigment. Add a bit of blue to yellow light and it looks white.
You can use a bit of harsher chemicals that remove stains more effectively but would risk removing the pigment in colored clothing
The PH level can differ and it can be higher for white fabric, to better remove stains.
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