Paints and inks have different coloring agents. Paints are made up of tiny bits of colored minerals like clays or talc suspended in a colorless matrix such as polyester or vinyl. A chemical union of tannic acid and iron sulfate dissolved in water or alcohol forms the coloring agent of some inks. Others use such dyes as aniline.
Ink soaks into the surface that it colors, but paints do not soak in -- they cover the surface with a coating. In fact, a primer is usually applied to the surface before painting so that the paint won't soak in. For this reason, paint sometimes chips off the surface that it covers, but this never happens in the case of ink.
Paint not only colors a surface, but also provides the surface with a protective covering that will keep it from deteriorating when assailed by such agents as wind and rain. Ink offers no such protection.