When the reflection color is white = colorless = the light that hits the surface will be reflected with the same color. So if a red ball is reflected in a blue mug with white reflection, then the reflected ball will be sill red.
When the reflection color is non-white = colored = the light that hits the surface will be filtered.
In reality if there is a single material, it always (almost always?) has colorless reflection.
If a reflection is colored, it is usually a thin layer of some kind of colored medium. Imagine white plastic covered with a very thin layer of yellow transparent coating. The reflections on such objects will be yellow, but technically that is not a single material.