CID is an invisible tag that stores the unique ID of the scene object to leverage it in interactive rendering together with motion blur and optimize some dynamic object changes during edit.
This tag is automatically created for each object if you use CentiLeo. If you uninstall the plugin you will see these tags for all scene objects. And it's also visible in the list of options of tags - but this is a design of Cinema itself, not possible to remove it from that list.

This link shows how to use the vertex color tag: https://centileo.com/news/145/centileo-for-cinema4d-0610/ (see the bottom most picture)
While the vertex map tag can be used in a similar way. It's just a drawing on the surface of the object which stores the values in the vertex map tag. And the shader node for vertex map just gives you access to these values. You can reference up to 10 tags (and more in the new versions) that can be attached per object where tag id = 0 in the node reference the most left vertex map tag and so on.

Sometimes other plugins like Realflow use vertex maps to generate velocity values of deforming object. And these vertex map tags can be used by CentiLeo object tag to use deformation motion blur as described in this short example https://centileo.com/news/150/centileo-for-cinema-0615/

Vertex map has various usage. E.g. Realflow generates them for deformation motion blur as I described before.
You can also draw a vertex map for your object. And a tag will appear in the list of object tags.
You may draw several vertex maps and they will generate Vertex Map tag 0, Vertex Map tag 1, ... Vertex Map tag N, from left to right in the tag list of the object.

If you want to get vertex map values for your shaders you should just use the vertex map node and connect it to the field you like, for example to determine reflection color properties or material amount in the mix material. In the vertex map node you only choose the vertex map tag ID.

Same story with vertex color tag!

As for mograph tag then it's automatically created for Mograph objects, e.g. for Cloner with multi-instancing, for mograph matrix, etc. You only need to know how to manage the mograph to generate UVW and Color data inside this mograph tag. In CentiLeo this mograph info can be accessed using mograph info node.

Redshift uses the names of tags to link them, but CentiLeo and native Cinema 4D nodes use tag id from left to right. Thats all the difference

In vertex map node you only determine the tag ID, and then you assign all this material to one or more objects and for each object the speficied vertex map tag will be used if the tag is available.