The question would be, what objects, what colors (…or materials?)
What is the reason you would like to do that?
Here are some examples:
My favorite normally is the following. (…it always depends, a general question produces general answers.) I assume the Connect object might be an already known starting point (set the Phong value to something better, perhaps). Use a MoGraph Fracture as a parent on top of the Connect object. In the Fracture> Object Mode> Explode Segments & Connect. Example here CV3_S24_drs_21_MGfr_01.c4d
If you use the Object Manager> Object> Connect and Delete to great the group, while each object has the UVs on its own tile, then the Variation Shader might work. I guess that is not often the case, But I had to mention it. Example: CV3_S24_drs_21_TXvr_01.c4d
With another Connect and Delete approach, while each previous single object has a Polygon selection with its own name, then the Material Tags can supply the color. Example CV3_S24_drs_21_TXsl_01.c4d
If all objects are visible from one point (no object obscures another), then a single painted texture in a Flat project could be used. Connect and Delete was used to create example CV3_S24_drs_21_TXf0_01.c4d
A variant of the first suggestion would be to use instead of the Connect Object the Connect and Delete option. Note that the Connect Object is a generator and has to recreate the setup again for each frame. This could pretty much slow down the machine if the objects produce a significant amount of Polygons. Example: CV3_S24_drs_21_TXfc_01.c4d
As a side note, you could work with Vertex Colors, but I would need a scene file to see if that could work for your project. The Example CV3_S24_drs_21_TXvc_01.c4d Was done by dragging the setup of CV3_S24_drs_21_TXfc_01.c4d into the Vertex Color, then delete it after the Freeze had stored the colors.
All the best