Clean out dryer lint.
Thus were you to load the washer with clothes of only one color-say a uniform purple-the lint would indeed be of that color.
Be sure not to use either paper towels, which can leave lint and other fibers, or solutions like window cleaner, which can degrade your display.
In an idealized situation, where the threads contained all the primary colors in equal amounts, and were not obscured by dirt and other residues, the lint would appear to be white, not gray.
That lint you see is from your clothes slowly being disintegrated by all the tumbling.
Her hand brushed a speck of lint from her skirt.