Can pigment ink somehow be used like a stamp-pad?
Based on your description and on research based on the terms you've provided, I expect that you have what users would consider an ordinary colored rubber stamp pad. Unfortunately, most of your description leads me to believe that it has deteriorated due to age or exposure to air.

The pad should be firm sponge feel and slightly moist. If it had dried and deteriorated, it will behave in a manner similar to an aged kitchen sponge.

The fragments you've described sound like "chunks" of the pad breaking away. As they are typically well impregnated with pigment, when you are experiencing the smearing, it is due to the higher concentration of pigment being released from the fragmenting fragment.